diff --git a/.github/workflows/release.yml b/.github/workflows/release.yml
index c55a742ee..a03dc8423 100644
--- a/.github/workflows/release.yml
+++ b/.github/workflows/release.yml
@@ -51,7 +51,7 @@ jobs:
     - name: Set up Python
       uses: actions/setup-python@v1
       with:
-        python-version: 3.8
+        python-version: 3.9
     - name: Give PyPI some time to update the index
       run: sleep 240
     - name: Attempt install from PyPI
diff --git a/.github/workflows/test.yml b/.github/workflows/test.yml
index be5624169..d56c8ade1 100644
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -24,7 +24,7 @@ jobs:
       - style
     strategy:
       matrix:
-        python-version: [ '3.8', '3.10']
+        python-version: [ '3.9', '3.11']
     steps:
     - uses: actions/checkout@v1
     - name: Set up Python ${{ matrix.python-version }}
diff --git a/.readthedocs.yaml b/.readthedocs.yaml
index cb5f162c6..611a9bebb 100644
--- a/.readthedocs.yaml
+++ b/.readthedocs.yaml
@@ -1,7 +1,7 @@
 version: 2
 
 python:
-  version: "3.8"
+  version: "3.9"
   install:
       - method: pip
         path: .
diff --git a/blackjax/adaptation/mass_matrix.py b/blackjax/adaptation/mass_matrix.py
index 4cd84492e..dc0730161 100644
--- a/blackjax/adaptation/mass_matrix.py
+++ b/blackjax/adaptation/mass_matrix.py
@@ -18,7 +18,7 @@
 parameters used in Hamiltonian Monte Carlo.
 
 """
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -68,7 +68,7 @@ class MassMatrixAdaptationState(NamedTuple):
 
 def mass_matrix_adaptation(
     is_diagonal_matrix: bool = True,
-) -> Tuple[Callable, Callable, Callable]:
+) -> tuple[Callable, Callable, Callable]:
     """Adapts the values in the mass matrix by computing the covariance
     between parameters.
 
@@ -156,7 +156,7 @@ def final(mm_state: MassMatrixAdaptationState) -> MassMatrixAdaptationState:
     return init, update, final
 
 
-def welford_algorithm(is_diagonal_matrix: bool) -> Tuple[Callable, Callable, Callable]:
+def welford_algorithm(is_diagonal_matrix: bool) -> tuple[Callable, Callable, Callable]:
     r"""Welford's online estimator of covariance.
 
     It is possible to compute the variance of a population of values in an
@@ -231,7 +231,7 @@ def update(
 
     def final(
         wa_state: WelfordAlgorithmState,
-    ) -> Tuple[Array, int, Array]:
+    ) -> tuple[Array, int, Array]:
         mean, m2, sample_size = wa_state
         covariance = m2 / (sample_size - 1)
         return covariance, sample_size, mean
diff --git a/blackjax/adaptation/pathfinder_adaptation.py b/blackjax/adaptation/pathfinder_adaptation.py
index 3d05bc1d1..c70ed3f99 100644
--- a/blackjax/adaptation/pathfinder_adaptation.py
+++ b/blackjax/adaptation/pathfinder_adaptation.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Implementation of the Pathinder warmup for the HMC family of sampling algorithms."""
-from typing import Callable, NamedTuple, Tuple, Union
+from typing import Callable, NamedTuple, Union
 
 import jax
 import jax.numpy as jnp
@@ -128,7 +128,7 @@ def update(
             new_ss_state, new_step_size, adaptation_state.inverse_mass_matrix
         )
 
-    def final(warmup_state: PathfinderAdaptationState) -> Tuple[float, Array]:
+    def final(warmup_state: PathfinderAdaptationState) -> tuple[float, Array]:
         """Return the final values for the step size and inverse mass matrix."""
         step_size = jnp.exp(warmup_state.ss_state.log_step_size_avg)
         inverse_mass_matrix = warmup_state.inverse_mass_matrix
diff --git a/blackjax/adaptation/step_size.py b/blackjax/adaptation/step_size.py
index 298f702f6..2d6b0182f 100644
--- a/blackjax/adaptation/step_size.py
+++ b/blackjax/adaptation/step_size.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Step size adaptation"""
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -64,7 +64,7 @@ class DualAveragingAdaptationState(NamedTuple):
 
 def dual_averaging_adaptation(
     target: float, t0: int = 10, gamma: float = 0.05, kappa: float = 0.75
-) -> Tuple[Callable, Callable, Callable]:
+) -> tuple[Callable, Callable, Callable]:
     """Tune the step size in order to achieve a desired target acceptance rate.
 
     Let us note :math:`\\epsilon` the current step size, :math:`\\alpha_t` the
diff --git a/blackjax/adaptation/window_adaptation.py b/blackjax/adaptation/window_adaptation.py
index b99d787f1..cc871b4b6 100644
--- a/blackjax/adaptation/window_adaptation.py
+++ b/blackjax/adaptation/window_adaptation.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Implementation of the Stan warmup for the HMC family of sampling algorithms."""
-from typing import Callable, List, NamedTuple, Tuple, Union
+from typing import Callable, NamedTuple, Union
 
 import jax
 import jax.numpy as jnp
@@ -45,7 +45,7 @@ class WindowAdaptationState(NamedTuple):
 def base(
     is_mass_matrix_diagonal: bool,
     target_acceptance_rate: float = 0.80,
-) -> Tuple[Callable, Callable, Callable]:
+) -> tuple[Callable, Callable, Callable]:
     """Warmup scheme for sampling procedures based on euclidean manifold HMC.
     The schedule and algorithms used match Stan's :cite:p:`stan_hmc_param` as closely as possible.
 
@@ -191,7 +191,7 @@ def slow_final(warmup_state: WindowAdaptationState) -> WindowAdaptationState:
 
     def update(
         adaptation_state: WindowAdaptationState,
-        adaptation_stage: Tuple,
+        adaptation_stage: tuple,
         position: ArrayLikeTree,
         acceptance_rate: float,
     ) -> WindowAdaptationState:
@@ -233,7 +233,7 @@ def update(
 
         return warmup_state
 
-    def final(warmup_state: WindowAdaptationState) -> Tuple[float, Array]:
+    def final(warmup_state: WindowAdaptationState) -> tuple[float, Array]:
         """Return the final values for the step size and mass matrix."""
         step_size = jnp.exp(warmup_state.ss_state.log_step_size_avg)
         inverse_mass_matrix = warmup_state.imm_state.inverse_mass_matrix
@@ -362,7 +362,7 @@ def build_schedule(
     initial_buffer_size: int = 75,
     final_buffer_size: int = 50,
     first_window_size: int = 25,
-) -> List[Tuple[int, bool]]:
+) -> list[tuple[int, bool]]:
     """Return the schedule for Stan's warmup.
 
     The schedule below is intended to be as close as possible to Stan's :cite:p:`stan_hmc_param`.
diff --git a/blackjax/base.py b/blackjax/base.py
index 0ad6a1628..7f709b895 100644
--- a/blackjax/base.py
+++ b/blackjax/base.py
@@ -10,7 +10,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 from typing_extensions import Protocol
 
@@ -64,7 +64,7 @@ class UpdateFn(Protocol):
 
     """
 
-    def __call__(self, rng_key: PRNGKey, state: State) -> Tuple[State, Info]:
+    def __call__(self, rng_key: PRNGKey, state: State) -> tuple[State, Info]:
         """Update the current state using the sampling algorithm.
 
         Parameters
diff --git a/blackjax/mcmc/elliptical_slice.py b/blackjax/mcmc/elliptical_slice.py
index e8010ffb5..4ff310445 100644
--- a/blackjax/mcmc/elliptical_slice.py
+++ b/blackjax/mcmc/elliptical_slice.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Public API for the Elliptical Slice sampling Kernel"""
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -110,7 +110,7 @@ def kernel(
         rng_key: PRNGKey,
         state: EllipSliceState,
         logdensity_fn: Callable,
-    ) -> Tuple[EllipSliceState, EllipSliceInfo]:
+    ) -> tuple[EllipSliceState, EllipSliceInfo]:
         proposal_generator = elliptical_proposal(
             logdensity_fn, momentum_generator, mean
         )
@@ -205,7 +205,7 @@ def elliptical_proposal(
 
     def generate(
         rng_key: PRNGKey, state: EllipSliceState
-    ) -> Tuple[EllipSliceState, EllipSliceInfo]:
+    ) -> tuple[EllipSliceState, EllipSliceInfo]:
         position, logdensity = state
         key_momentum, key_uniform, key_theta = jax.random.split(rng_key, 3)
         # step 1: sample momentum
diff --git a/blackjax/mcmc/ghmc.py b/blackjax/mcmc/ghmc.py
index 62462ae68..a068acee7 100644
--- a/blackjax/mcmc/ghmc.py
+++ b/blackjax/mcmc/ghmc.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Public API for the Generalized (Non-reversible w/ persistent momentum) HMC Kernel"""
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -104,7 +104,7 @@ def kernel(
         momentum_inverse_scale: ArrayLikeTree,
         alpha: float,
         delta: float,
-    ) -> Tuple[GHMCState, hmc.HMCInfo]:
+    ) -> tuple[GHMCState, hmc.HMCInfo]:
         """Generate new sample with the Generalized HMC kernel.
 
         Parameters
diff --git a/blackjax/mcmc/hmc.py b/blackjax/mcmc/hmc.py
index ecdf2394e..228fd0b51 100644
--- a/blackjax/mcmc/hmc.py
+++ b/blackjax/mcmc/hmc.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Public API for the HMC Kernel"""
-from typing import Callable, NamedTuple, Tuple, Union
+from typing import Callable, NamedTuple, Union
 
 import jax
 
@@ -112,7 +112,7 @@ def kernel(
         step_size: float,
         inverse_mass_matrix: Array,
         num_integration_steps: int,
-    ) -> Tuple[HMCState, HMCInfo]:
+    ) -> tuple[HMCState, HMCInfo]:
         """Generate a new sample with the HMC kernel."""
 
         momentum_generator, kinetic_energy_fn, _ = metrics.gaussian_euclidean(
@@ -281,7 +281,7 @@ def hmc_proposal(
 
     def generate(
         rng_key, state: integrators.IntegratorState
-    ) -> Tuple[integrators.IntegratorState, HMCInfo]:
+    ) -> tuple[integrators.IntegratorState, HMCInfo]:
         """Generate a new chain state."""
         end_state = build_trajectory(state, step_size, num_integration_steps)
         end_state = flip_momentum(end_state)
diff --git a/blackjax/mcmc/mala.py b/blackjax/mcmc/mala.py
index 7e76202a0..0f4295a0e 100644
--- a/blackjax/mcmc/mala.py
+++ b/blackjax/mcmc/mala.py
@@ -13,7 +13,7 @@
 # limitations under the License.
 """Public API for Metropolis Adjusted Langevin kernels."""
 import operator
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -96,7 +96,7 @@ def transition_energy(state, new_state, step_size):
 
     def kernel(
         rng_key: PRNGKey, state: MALAState, logdensity_fn: Callable, step_size: float
-    ) -> Tuple[MALAState, MALAInfo]:
+    ) -> tuple[MALAState, MALAInfo]:
         """Generate a new sample with the MALA kernel."""
         grad_fn = jax.value_and_grad(logdensity_fn)
         integrator = diffusions.overdamped_langevin(grad_fn)
diff --git a/blackjax/mcmc/metrics.py b/blackjax/mcmc/metrics.py
index 262345abd..a24bc00b4 100644
--- a/blackjax/mcmc/metrics.py
+++ b/blackjax/mcmc/metrics.py
@@ -27,7 +27,7 @@
 We can also generate a relativistic dynamic :cite:p:`lu2017relativistic`.
 
 """
-from typing import Callable, Tuple
+from typing import Callable
 
 import jax.numpy as jnp
 import jax.scipy as jscipy
@@ -43,7 +43,7 @@
 
 def gaussian_euclidean(
     inverse_mass_matrix: Array,
-) -> Tuple[Callable, EuclideanKineticEnergy, Callable]:
+) -> tuple[Callable, EuclideanKineticEnergy, Callable]:
     r"""Hamiltonian dynamic on euclidean manifold with normally-distributed momentum :cite:p:`betancourt2013general`.
 
     The gaussian euclidean metric is a euclidean metric further characterized
diff --git a/blackjax/mcmc/nuts.py b/blackjax/mcmc/nuts.py
index 6f3b4fc4b..e09841ccf 100644
--- a/blackjax/mcmc/nuts.py
+++ b/blackjax/mcmc/nuts.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Public API for the NUTS Kernel"""
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -121,7 +121,7 @@ def kernel(
         logdensity_fn: Callable,
         step_size: float,
         inverse_mass_matrix: Array,
-    ) -> Tuple[hmc.HMCState, NUTSInfo]:
+    ) -> tuple[hmc.HMCState, NUTSInfo]:
         """Generate a new sample with the NUTS kernel."""
 
         (
diff --git a/blackjax/mcmc/periodic_orbital.py b/blackjax/mcmc/periodic_orbital.py
index ae0d9fede..a8bb54787 100644
--- a/blackjax/mcmc/periodic_orbital.py
+++ b/blackjax/mcmc/periodic_orbital.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 """Public API for Periodic Orbital Kernel"""
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -142,7 +142,7 @@ def kernel(
         step_size: float,
         inverse_mass_matrix: Array,
         period: int,
-    ) -> Tuple[PeriodicOrbitalState, PeriodicOrbitalInfo]:
+    ) -> tuple[PeriodicOrbitalState, PeriodicOrbitalInfo]:
         """Generate a new orbit with the Periodic Orbital kernel.
 
         Choose a step from the orbit with probability proportional to its weights.
@@ -325,7 +325,7 @@ def periodic_orbital_proposal(
 
     def generate(
         direction: int, init_state: integrators.IntegratorState
-    ) -> Tuple[PeriodicOrbitalState, PeriodicOrbitalInfo]:
+    ) -> tuple[PeriodicOrbitalState, PeriodicOrbitalInfo]:
         """Generate orbit by applying bijection forwards and backwards on period.
 
         As described in algorithm 2 of :cite:p:`neklyudov2022orbital`, each iteration of the periodic orbital
diff --git a/blackjax/mcmc/proposal.py b/blackjax/mcmc/proposal.py
index 19983a450..642c76623 100644
--- a/blackjax/mcmc/proposal.py
+++ b/blackjax/mcmc/proposal.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -42,7 +42,7 @@ class Proposal(NamedTuple):
 
 def proposal_generator(
     energy: Callable, divergence_threshold: float
-) -> Tuple[Callable, Callable]:
+) -> tuple[Callable, Callable]:
     """
 
     Parameters
@@ -61,7 +61,7 @@ def proposal_generator(
     def new(state: TrajectoryState) -> Proposal:
         return Proposal(state, energy(state), 0.0, -jnp.inf)
 
-    def update(initial_energy: float, state: TrajectoryState) -> Tuple[Proposal, bool]:
+    def update(initial_energy: float, state: TrajectoryState) -> tuple[Proposal, bool]:
         """Generate a new proposal from a trajectory state.
 
         The trajectory state records information about the position in the state
@@ -95,7 +95,7 @@ def proposal_from_energy_diff(
     new_energy: float,
     divergence_threshold: float,
     state: TrajectoryState,
-) -> Tuple[Proposal, bool]:
+) -> tuple[Proposal, bool]:
     """Computes a new proposal from the energy difference between two states.
 
     It also verifies whether this difference is a divergence, if the
@@ -141,7 +141,7 @@ def asymmetric_proposal_generator(
     transition_energy_fn: Callable,
     divergence_threshold: float,
     proposal_factory: Callable = proposal_from_energy_diff,
-) -> Tuple[Callable, Callable]:
+) -> tuple[Callable, Callable]:
     """A proposal generator that takes into account the transition between
     two states to compute a new proposal.
 
@@ -171,7 +171,7 @@ def update(
         initial_state: TrajectoryState,
         state: TrajectoryState,
         **energy_params,
-    ) -> Tuple[Proposal, bool]:
+    ) -> tuple[Proposal, bool]:
         new_energy = transition_energy_fn(initial_state, state, **energy_params)
         prev_energy = transition_energy_fn(state, initial_state, **energy_params)
         return proposal_factory(prev_energy, new_energy, divergence_threshold, state)
diff --git a/blackjax/mcmc/random_walk.py b/blackjax/mcmc/random_walk.py
index 3f260ffa4..6d97c7c08 100644
--- a/blackjax/mcmc/random_walk.py
+++ b/blackjax/mcmc/random_walk.py
@@ -60,7 +60,7 @@
         new_state, info = step(rng_key, state)
 
 """
-from typing import Callable, NamedTuple, Optional, Tuple
+from typing import Callable, NamedTuple, Optional
 
 import jax
 import numpy as np
@@ -171,7 +171,7 @@ def build_additive_step():
 
     def kernel(
         rng_key: PRNGKey, state: RWState, logdensity_fn: Callable, random_step: Callable
-    ) -> Tuple[RWState, RWInfo]:
+    ) -> tuple[RWState, RWInfo]:
         def proposal_generator(key_proposal, position):
             move_proposal = random_step(key_proposal, position)
             new_position = jax.tree_util.tree_map(jnp.add, position, move_proposal)
@@ -271,7 +271,7 @@ def kernel(
         state: RWState,
         logdensity_fn: Callable,
         proposal_distribution: Callable,
-    ) -> Tuple[RWState, RWInfo]:
+    ) -> tuple[RWState, RWInfo]:
         """
 
         Parameters
@@ -362,7 +362,7 @@ def kernel(
         logdensity_fn: Callable,
         transition_generator: Callable,
         proposal_logdensity_fn: Optional[Callable] = None,
-    ) -> Tuple[RWState, RWInfo]:
+    ) -> tuple[RWState, RWInfo]:
         """Move the chain by one step using the Rosenbluth Metropolis Hastings
         algorithm.
 
@@ -493,7 +493,7 @@ def build_trajectory(rng_key, initial_state: RWState) -> RWState:
         new_position = transition_distribution(rng_key, position)
         return RWState(new_position, logdensity_fn(new_position))
 
-    def generate(rng_key, state: RWState) -> Tuple[RWState, bool, float]:
+    def generate(rng_key, state: RWState) -> tuple[RWState, bool, float]:
         key_proposal, key_accept = jax.random.split(rng_key, 2)
         end_state = build_trajectory(key_proposal, state)
         new_proposal, _ = generate_proposal(state, end_state)
diff --git a/blackjax/mcmc/trajectory.py b/blackjax/mcmc/trajectory.py
index 265afc351..81d369c0b 100644
--- a/blackjax/mcmc/trajectory.py
+++ b/blackjax/mcmc/trajectory.py
@@ -36,7 +36,7 @@
 memory by keeping states that will subsequently be discarded.
 
 """
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -70,7 +70,7 @@ def append_to_trajectory(trajectory: Trajectory, state: IntegratorState) -> Traj
 
 def reorder_trajectories(
     direction: int, trajectory: Trajectory, new_trajectory: Trajectory
-) -> Tuple[Trajectory, Trajectory]:
+) -> tuple[Trajectory, Trajectory]:
     """Order the two trajectories depending on the direction."""
     return jax.lax.cond(
         direction > 0,
diff --git a/blackjax/optimizers/dual_averaging.py b/blackjax/optimizers/dual_averaging.py
index 1e3eca782..94b7aaa34 100644
--- a/blackjax/optimizers/dual_averaging.py
+++ b/blackjax/optimizers/dual_averaging.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax.numpy as jnp
 
@@ -52,7 +52,7 @@ class DualAveragingState(NamedTuple):
 
 def dual_averaging(
     t0: int = 10, gamma: float = 0.05, kappa: float = 0.75
-) -> Tuple[Callable, Callable, Callable]:
+) -> tuple[Callable, Callable, Callable]:
     """Find the state that minimizes an objective function using a primal-dual
     subgradient method.
 
diff --git a/blackjax/optimizers/lbfgs.py b/blackjax/optimizers/lbfgs.py
index ba688ff8e..de653108c 100644
--- a/blackjax/optimizers/lbfgs.py
+++ b/blackjax/optimizers/lbfgs.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -70,7 +70,7 @@ def minimize_lbfgs(
     gtol: float = 1e-08,
     ftol: float = 1e-05,
     maxls: int = 1000,
-) -> Tuple[OptStep, LBFGSHistory]:
+) -> tuple[OptStep, LBFGSHistory]:
     """
     Minimize a function using L-BFGS
 
@@ -152,7 +152,7 @@ def _minimize_lbfgs(
     gtol: float,
     ftol: float,
     maxls: int,
-) -> Tuple[OptStep, LBFGSHistory]:
+) -> tuple[OptStep, LBFGSHistory]:
     def lbfgs_one_step(carry, i):
         (params, state), previous_history = carry
 
diff --git a/blackjax/smc/adaptive_tempered.py b/blackjax/smc/adaptive_tempered.py
index d2b24a9f7..bbb71761a 100644
--- a/blackjax/smc/adaptive_tempered.py
+++ b/blackjax/smc/adaptive_tempered.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, Dict, Tuple
+from typing import Callable
 
 import jax
 import jax.numpy as jnp
@@ -95,7 +95,7 @@ def kernel(
         state: tempered.TemperedSMCState,
         num_mcmc_steps: int,
         mcmc_parameters: dict,
-    ) -> Tuple[tempered.TemperedSMCState, base.SMCInfo]:
+    ) -> tuple[tempered.TemperedSMCState, base.SMCInfo]:
         delta = compute_delta(state)
         lmbda = delta + state.lmbda
         return tempered_kernel(rng_key, state, num_mcmc_steps, lmbda, mcmc_parameters)
@@ -143,7 +143,7 @@ def __new__(  # type: ignore[misc]
         loglikelihood_fn: Callable,
         mcmc_step_fn: Callable,
         mcmc_init_fn: Callable,
-        mcmc_parameters: Dict,
+        mcmc_parameters: dict,
         resampling_fn: Callable,
         target_ess: float,
         root_solver: Callable = solver.dichotomy,
diff --git a/blackjax/smc/base.py b/blackjax/smc/base.py
index 0732e0404..52d1338ef 100644
--- a/blackjax/smc/base.py
+++ b/blackjax/smc/base.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Optional, Tuple
+from typing import Callable, NamedTuple, Optional
 
 import jax
 import jax.numpy as jnp
@@ -59,7 +59,7 @@ def step(
     weigh_fn: Callable,
     resample_fn: Callable,
     num_resampled: Optional[int] = None,
-) -> Tuple[SMCState, SMCInfo]:
+) -> tuple[SMCState, SMCInfo]:
     """General SMC sampling step.
 
     `update_fn` here corresponds to the Markov kernel $M_{t+1}$, and `weigh_fn`
diff --git a/blackjax/smc/tempered.py b/blackjax/smc/tempered.py
index f7de5768f..40e95b665 100644
--- a/blackjax/smc/tempered.py
+++ b/blackjax/smc/tempered.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, Dict, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -97,7 +97,7 @@ def kernel(
         num_mcmc_steps: int,
         lmbda: float,
         mcmc_parameters: dict,
-    ) -> Tuple[TemperedSMCState, smc.base.SMCInfo]:
+    ) -> tuple[TemperedSMCState, smc.base.SMCInfo]:
         """Move the particles one step using the Tempered SMC algorithm.
 
         Parameters
@@ -191,7 +191,7 @@ def __new__(  # type: ignore[misc]
         loglikelihood_fn: Callable,
         mcmc_step_fn: Callable,
         mcmc_init_fn: Callable,
-        mcmc_parameters: Dict,
+        mcmc_parameters: dict,
         resampling_fn: Callable,
         num_mcmc_steps: int = 10,
     ) -> SamplingAlgorithm:
diff --git a/blackjax/vi/meanfield_vi.py b/blackjax/vi/meanfield_vi.py
index e7f5c409d..8d5defa15 100644
--- a/blackjax/vi/meanfield_vi.py
+++ b/blackjax/vi/meanfield_vi.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Tuple
+from typing import Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -61,7 +61,7 @@ def step(
     optimizer: GradientTransformation,
     num_samples: int = 5,
     stl_estimator: bool = True,
-) -> Tuple[MFVIState, MFVIInfo]:
+) -> tuple[MFVIState, MFVIInfo]:
     """Approximate the target density using the mean-field approximation.
 
     Parameters
@@ -141,7 +141,7 @@ def __new__(
         def init_fn(position: ArrayLikeTree):
             return cls.init(position, optimizer)
 
-        def step_fn(rng_key: PRNGKey, state: MFVIState) -> Tuple[MFVIState, MFVIInfo]:
+        def step_fn(rng_key: PRNGKey, state: MFVIState) -> tuple[MFVIState, MFVIInfo]:
             return cls.step(rng_key, state, logdensity_fn, optimizer, num_samples)
 
         def sample_fn(rng_key: PRNGKey, state: MFVIState, num_samples: int):
diff --git a/blackjax/vi/pathfinder.py b/blackjax/vi/pathfinder.py
index 7cb40e437..e494b026c 100644
--- a/blackjax/vi/pathfinder.py
+++ b/blackjax/vi/pathfinder.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import Callable, NamedTuple, Tuple, Union
+from typing import Callable, NamedTuple, Union
 
 import jax
 import jax.numpy as jnp
@@ -79,7 +79,7 @@ def approximate(
     maxls=1000,
     gtol=1e-08,
     ftol=1e-05,
-) -> Tuple[PathfinderState, PathfinderInfo]:
+) -> tuple[PathfinderState, PathfinderInfo]:
     """Pathfinder variational inference algorithm.
 
     Pathfinder locates normal approximations to the target density along a
@@ -200,7 +200,7 @@ def path_finder_body_fn(rng_key, S, Z, alpha_l, theta, theta_grad):
 def sample(
     rng_key: PRNGKey,
     state: PathfinderState,
-    num_samples: Union[int, Tuple[()], Tuple[int]] = (),
+    num_samples: Union[int, tuple[()], tuple[int]] = (),
 ) -> ArrayTree:
     """Draw from the Pathfinder approximation of the target distribution.
 
diff --git a/blackjax/vi/svgd.py b/blackjax/vi/svgd.py
index 9ec7f28ff..f93941aee 100644
--- a/blackjax/vi/svgd.py
+++ b/blackjax/vi/svgd.py
@@ -1,5 +1,5 @@
 import functools
-from typing import Any, Callable, Dict, NamedTuple
+from typing import Any, Callable, NamedTuple
 
 import jax
 import jax.numpy as jnp
@@ -14,13 +14,13 @@
 
 class SVGDState(NamedTuple):
     particles: ArrayTree
-    kernel_parameters: Dict[str, ArrayTree]
+    kernel_parameters: dict[str, ArrayTree]
     opt_state: Any
 
 
 def init(
     initial_particles: ArrayLikeTree,
-    kernel_parameters: Dict[str, Any],
+    kernel_parameters: dict[str, Any],
     optimizer: optax.GradientTransformation,
 ) -> SVGDState:
     """
@@ -156,7 +156,7 @@ def __new__(
 
         def init_fn(
             initial_position: ArrayLikeTree,
-            kernel_parameters: Dict[str, Any] = {"length_scale": 1.0},
+            kernel_parameters: dict[str, Any] = {"length_scale": 1.0},
         ):
             return cls.init(initial_position, kernel_parameters, optimizer)
 
diff --git a/pyproject.toml b/pyproject.toml
index 51ac3b636..ae630a9ab 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -6,7 +6,7 @@ build-backend = "setuptools.build_meta"
 name = "blackjax"
 authors= [{name = "The Blackjax team", email = "remi@thetypicalset.com"}]
 description = "Flexible and fast sampling in Python"
-requires-python = ">=3.8"
+requires-python = ">=3.9"
 keywords=[
     "probability",
     "machine learning",
@@ -22,9 +22,9 @@ classifiers = [
     "License :: OSI Approved :: Apache Software License",
     "Operating System :: MacOS",
     "Operating System :: POSIX",
-    "Programming Language :: Python :: 3.8",
     "Programming Language :: Python :: 3.9",
     "Programming Language :: Python :: 3.10",
+    "Programming Language :: Python :: 3.11",
     "Topic :: Education",
     "Topic :: Scientific/Engineering",
     "Topic :: Scientific/Engineering :: Artificial Intelligence",