diff --git a/pyquil/quilatom.py b/pyquil/quilatom.py
index 86fdec8f6..c3cfe5e59 100644
--- a/pyquil/quilatom.py
+++ b/pyquil/quilatom.py
@@ -25,19 +25,19 @@ class QuilAtom(object):
     Abstract class for atomic elements of Quil.
     """
 
-    def out(self):
+    def out(self) -> str:
         raise NotImplementedError()
 
-    def __str__(self):
+    def __str__(self) -> str:
         raise NotImplementedError()
 
-    def __eq__(self, other):
+    def __eq__(self, other: object) -> bool:
         raise NotImplementedError()
 
     def __ne__(self, other: object) -> bool:
         return not self.__eq__(other)
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         raise NotImplementedError()
 
 
@@ -70,7 +70,7 @@ def __eq__(self, other: object) -> bool:
 
 
 class QubitPlaceholder(QuilAtom):
-    def out(self):
+    def out(self) -> str:
         raise RuntimeError("Qubit {} has not been assigned an index".format(self))
 
     def __str__(self) -> str:
@@ -125,8 +125,11 @@ def unpack_qubit(qubit: QubitDesignator) -> Union[Qubit, QubitPlaceholder]:
         raise TypeError("qubit should be an int or Qubit instance")
 
 
-# Like the Tuple, the List must be length 2, where the first item is a string and the second an int.
-MemoryReferenceDesignator = Union['MemoryReference', Tuple[str, int], List, str]
+# Like the Tuple, the List must be length 2, where the first item is a string and the second an
+# int. However, specifying Union[str, int] as the generic type argument to List doesn't sufficiently
+# constrain the types, and mypy gets confused in unpack_classical_reg, below. Hence, just specify
+# List here and add a "# type: ignore" comment to silence mypy --strict.
+MemoryReferenceDesignator = Union['MemoryReference', Tuple[str, int], List, str]  # type: ignore
 
 
 def unpack_classical_reg(c: MemoryReferenceDesignator) -> 'MemoryReference':
@@ -184,7 +187,7 @@ class LabelPlaceholder(QuilAtom):
     def __init__(self, prefix: str = "L") -> None:
         self.prefix = prefix
 
-    def out(self):
+    def out(self) -> str:
         raise RuntimeError("Label has not been assigned a name")
 
     def __str__(self) -> str:
@@ -245,7 +248,7 @@ def format_parameter(element: ParameterDesignator) -> str:
     assert False, "Invalid parameter: %r" % element
 
 
-ExpressionValue = Union[int, float]
+ExpressionValue = Union[int, float, complex]
 ExpressionOrValue = Union['Expression', ExpressionValue]
 
 
@@ -365,7 +368,8 @@ class Function(Expression):
     """
     Supported functions in Quil are sin, cos, sqrt, exp, and cis
     """
-    def __init__(self, name: str, expression: ExpressionOrValue, fn: Callable) -> None:
+    def __init__(self, name: str, expression: ExpressionOrValue,
+                 fn: Callable[[ExpressionValue], ExpressionValue]) -> None:
         self.name = name
         self.expression = expression
         self.fn = fn
@@ -381,7 +385,7 @@ def __eq__(self, other: object) -> bool:
                 and self.name == other.name
                 and self.expression == other.expression)
 
-    def __neq__(self, other: 'Function'):
+    def __neq__(self, other: 'Function') -> bool:
         return not self.__eq__(other)
 
 
@@ -402,7 +406,7 @@ def quil_exp(expression: ExpressionOrValue) -> Function:
 
 
 def quil_cis(expression: ExpressionOrValue) -> Function:
-    return Function('CIS', expression, lambda x: np.exp(1j * x))
+    return Function('CIS', expression, lambda x: np.exp(1j * x))  # type: ignore
 
 
 class BinaryExp(Expression):
@@ -411,7 +415,7 @@ class BinaryExp(Expression):
     associates: ClassVar[str]
 
     @staticmethod
-    def fn(a: ExpressionOrValue, b: ExpressionOrValue):
+    def fn(a: ExpressionOrValue, b: ExpressionOrValue) -> Union['BinaryExp', ExpressionValue]:
         raise NotImplementedError
 
     def __init__(self, op1: ExpressionOrValue, op2: ExpressionOrValue) -> None:
diff --git a/pyquil/quilbase.py b/pyquil/quilbase.py
index b81c487aa..12764fc5a 100644
--- a/pyquil/quilbase.py
+++ b/pyquil/quilbase.py
@@ -89,7 +89,7 @@ class Gate(AbstractInstruction):
     This is the pyQuil object for a quantum gate instruction.
     """
 
-    def __init__(self, name, params, qubits):
+    def __init__(self, name, params, qubits) -> None:
         if not isinstance(name, string_types):
             raise TypeError("Gate name must be a string")
 
@@ -221,7 +221,7 @@ class Measurement(AbstractInstruction):
     This is the pyQuil object for a Quil measurement instruction.
     """
 
-    def __init__(self, qubit, classical_reg):
+    def __init__(self, qubit, classical_reg) -> None:
         if not isinstance(qubit, (Qubit, QubitPlaceholder)):
             raise TypeError("qubit should be a Qubit")
         if classical_reg and not isinstance(classical_reg, MemoryReference):
@@ -251,7 +251,7 @@ class ResetQubit(AbstractInstruction):
     This is the pyQuil object for a Quil targeted reset instruction.
     """
 
-    def __init__(self, qubit):
+    def __init__(self, qubit) -> None:
         if not isinstance(qubit, (Qubit, QubitPlaceholder)):
             raise TypeError("qubit should be a Qubit")
         self.qubit = qubit
@@ -275,7 +275,7 @@ class DefGate(AbstractInstruction):
     :param list parameters: list of parameters that are used in this gate
     """
 
-    def __init__(self, name, matrix, parameters=None):
+    def __init__(self, name, matrix, parameters=None) -> None:
         if not isinstance(name, string_types):
             raise TypeError("Gate name must be a string")
 
@@ -370,7 +370,7 @@ def num_args(self):
 
 
 class DefPermutationGate(DefGate):
-    def __init__(self, name, permutation):
+    def __init__(self, name, permutation) -> None:
         if not isinstance(name, string_types):
             raise TypeError("Gate name must be a string")
 
@@ -410,7 +410,7 @@ class JumpTarget(AbstractInstruction):
     Representation of a target that can be jumped to.
     """
 
-    def __init__(self, label):
+    def __init__(self, label) -> None:
         if not isinstance(label, (Label, LabelPlaceholder)):
             raise TypeError("label must be a Label")
         self.label = label
@@ -428,7 +428,7 @@ class JumpConditional(AbstractInstruction):
     """
     op = NotImplemented
 
-    def __init__(self, target, condition):
+    def __init__(self, target, condition) -> None:
         if not isinstance(target, (Label, LabelPlaceholder)):
             raise TypeError("target should be a Label")
         if not isinstance(condition, MemoryReference):
@@ -496,7 +496,7 @@ class UnaryClassicalInstruction(AbstractInstruction):
     The abstract class for unary classical instructions.
     """
 
-    def __init__(self, target):
+    def __init__(self, target) -> None:
         if not isinstance(target, MemoryReference):
             raise TypeError("target operand should be an MemoryReference")
         self.target = target
@@ -525,7 +525,7 @@ class LogicalBinaryOp(AbstractInstruction):
     """
     op = NotImplemented
 
-    def __init__(self, left, right):
+    def __init__(self, left, right) -> None:
         if not isinstance(left, MemoryReference):
             raise TypeError("left operand should be an MemoryReference")
         if not isinstance(right, MemoryReference) and not isinstance(right, int):
@@ -570,7 +570,7 @@ class ClassicalOr(ClassicalInclusiveOr):
     Deprecated class.
     """
 
-    def __init__(self, left, right):
+    def __init__(self, left, right) -> None:
         warn("ClassicalOr has been deprecated. Replacing with "
              "ClassicalInclusiveOr. Use ClassicalInclusiveOr instead. "
              "NOTE: The operands to ClassicalInclusiveOr are inverted from "
@@ -583,7 +583,7 @@ class ArithmeticBinaryOp(AbstractInstruction):
     The abstract class for binary arithmetic classical instructions.
     """
 
-    def __init__(self, left, right):
+    def __init__(self, left, right) -> None:
         if not isinstance(left, MemoryReference):
             raise TypeError("left operand should be an MemoryReference")
         if not isinstance(right, MemoryReference) and not isinstance(right, int) and not isinstance(right, float):
@@ -633,7 +633,7 @@ class ClassicalMove(AbstractInstruction):
     """
     op = "MOVE"
 
-    def __init__(self, left, right):
+    def __init__(self, left, right) -> None:
         if not isinstance(left, MemoryReference):
             raise TypeError("Left operand of MOVE should be an MemoryReference.  "
                             "Note that the order of the operands in pyQuil 2.0 has reversed from "
@@ -653,7 +653,7 @@ class ClassicalFalse(ClassicalMove):
     Deprecated class.
     """
 
-    def __init__(self, target):
+    def __init__(self, target) -> None:
         super().__init__(target, 0)
         warn("ClassicalFalse is deprecated in favor of ClassicalMove.")
 
@@ -663,7 +663,7 @@ class ClassicalTrue(ClassicalMove):
     Deprecated class.
     """
 
-    def __init__(self, target):
+    def __init__(self, target) -> None:
         super().__init__(target, 1)
         warn("ClassicalTrue is deprecated in favor of ClassicalMove.")
 
@@ -675,7 +675,7 @@ class ClassicalExchange(AbstractInstruction):
 
     op = "EXCHANGE"
 
-    def __init__(self, left, right):
+    def __init__(self, left, right) -> None:
         if not isinstance(left, MemoryReference):
             raise TypeError("left operand should be an MemoryReference")
         if not isinstance(right, MemoryReference):
@@ -694,7 +694,7 @@ class ClassicalConvert(AbstractInstruction):
 
     op = "CONVERT"
 
-    def __init__(self, left, right):
+    def __init__(self, left, right) -> None:
         if not isinstance(left, MemoryReference):
             raise TypeError("left operand should be an MemoryReference")
         if not isinstance(right, MemoryReference):
@@ -713,7 +713,7 @@ class ClassicalLoad(AbstractInstruction):
 
     op = "LOAD"
 
-    def __init__(self, target, left, right):
+    def __init__(self, target, left, right) -> None:
         if not isinstance(target, MemoryReference):
             raise TypeError("target operand should be an MemoryReference")
         if not isinstance(right, MemoryReference):
@@ -733,7 +733,7 @@ class ClassicalStore(AbstractInstruction):
 
     op = "STORE"
 
-    def __init__(self, target, left, right):
+    def __init__(self, target, left, right) -> None:
         if not isinstance(left, MemoryReference):
             raise TypeError("left operand should be an MemoryReference")
         if not (isinstance(right, MemoryReference) or isinstance(right, int)
@@ -752,7 +752,7 @@ class ClassicalComparison(AbstractInstruction):
     Abstract class for ternary comparison instructions.
     """
 
-    def __init__(self, target, left, right):
+    def __init__(self, target, left, right) -> None:
         if not isinstance(target, MemoryReference):
             raise TypeError("target operand should be an MemoryReference")
         if not isinstance(left, MemoryReference):
@@ -810,7 +810,7 @@ class Jump(AbstractInstruction):
     Representation of an unconditional jump instruction (JUMP).
     """
 
-    def __init__(self, target):
+    def __init__(self, target) -> None:
         if not isinstance(target, (Label, LabelPlaceholder)):
             raise TypeError("target should be a Label")
         self.target = target
@@ -829,7 +829,7 @@ class Pragma(AbstractInstruction):
 
     """
 
-    def __init__(self, command, args=(), freeform_string=""):
+    def __init__(self, command, args=(), freeform_string="") -> None:
         if not isinstance(command, string_types):
             raise TypeError("Pragma's require an identifier.")
 
@@ -871,7 +871,7 @@ class Declare(AbstractInstruction):
 
     """
 
-    def __init__(self, name, memory_type, memory_size=1, shared_region=None, offsets=None):
+    def __init__(self, name, memory_type, memory_size=1, shared_region=None, offsets=None) -> None:
         self.name = name
         self.memory_type = memory_type
         self.memory_size = memory_size
@@ -908,7 +908,7 @@ class RawInstr(AbstractInstruction):
     A raw instruction represented as a string.
     """
 
-    def __init__(self, instr_str):
+    def __init__(self, instr_str) -> None:
         if not isinstance(instr_str, string_types):
             raise TypeError("Raw instructions require a string.")
         self.instr = instr_str