diff --git a/pytorch_lightning/trainer/training_loop.py b/pytorch_lightning/trainer/training_loop.py
index 5b16ea5b1fb1a..7dbf0674ffa3b 100644
--- a/pytorch_lightning/trainer/training_loop.py
+++ b/pytorch_lightning/trainer/training_loop.py
@@ -192,12 +192,13 @@ def on_train_end(self):
 
     def check_checkpoint_callback(self, should_save, is_last=False):
         # TODO bake this logic into the checkpoint callback
-        if should_save:
-            checkpoint_callbacks = [c for c in self.trainer.callbacks if isinstance(c, ModelCheckpoint)]
-            if is_last and any(c.save_last for c in checkpoint_callbacks):
-                rank_zero_info('Saving latest checkpoint...')
-            model = self.trainer.get_model()
-            [c.on_validation_end(self.trainer, model) for c in checkpoint_callbacks]
+        if not should_save:
+            return
+        checkpoint_callbacks = [c for c in self.trainer.callbacks if isinstance(c, ModelCheckpoint)]
+        if is_last and any(c.save_last for c in checkpoint_callbacks):
+            rank_zero_info('Saving latest checkpoint...')
+        model = self.trainer.get_model()
+        [c.on_validation_end(self.trainer, model) for c in checkpoint_callbacks]
 
     def on_train_epoch_start(self, epoch):
         model = self.trainer.get_model()
@@ -589,8 +590,7 @@ def run_training_epoch(self):
         # epoch end hook
         self.run_on_epoch_end_hook()
 
-        # increment the global step once
-        # progress global step according to grads progress
+        # increment the global step once progress global step according to grads progress
         self.increment_accumulated_grad_global_step()
 
     def run_training_batch(self, batch, batch_idx, dataloader_idx):
diff --git a/tests/base/model_valid_steps.py b/tests/base/model_valid_steps.py
index 015368ab0e0fc..6e2ad247f84e3 100644
--- a/tests/base/model_valid_steps.py
+++ b/tests/base/model_valid_steps.py
@@ -2,6 +2,7 @@
 from collections import OrderedDict
 from pytorch_lightning.core.step_result import EvalResult
 
+import numpy as np
 import torch
 
 
@@ -35,17 +36,16 @@ def validation_step(self, batch, batch_idx, *args, **kwargs):
         return output
 
     def validation_step__decreasing(self, batch, batch_idx, *args, **kwargs):
-        if not hasattr(self, 'running_loss'):
-            self.running_loss = 1
-        if not hasattr(self, 'running_acc'):
-            self.running_acc = 0
+        if not hasattr(self, 'running'):
+            self.running = 0
+        self.running += 1
 
-        self.running_loss -= 1e-2
-        self.running_acc += 1e-2
+        running_loss = np.e ** (10 / self.running) - 1
+        running_acc = np.log(self.running + 1)
 
         output = OrderedDict({
-            'val_loss': torch.tensor(self.running_loss),
-            'val_acc': torch.tensor(self.running_acc),
+            'val_loss': torch.tensor(running_loss),
+            'val_acc': torch.tensor(running_acc),
         })
         return output
 
diff --git a/tests/callbacks/test_checkpoint_frequency.py b/tests/callbacks/test_checkpoint_frequency.py
index 356acff6d1216..b629f63a98698 100644
--- a/tests/callbacks/test_checkpoint_frequency.py
+++ b/tests/callbacks/test_checkpoint_frequency.py
@@ -66,8 +66,8 @@ def test_mc_called(tmpdir):
     assert len(trainer.dev_debugger.checkpoint_callback_history) == 0
 
 
-@pytest.mark.parametrize("period", [0.2, 0.5, 0.8, 1.])
-@pytest.mark.parametrize("epochs", [2, 5])
+@pytest.mark.parametrize("period", [0.2, 0.3, 0.5, 0.8, 1.])
+@pytest.mark.parametrize("epochs", [1, 2])
 def test_model_checkpoint_period(tmpdir, epochs, period):
     os.environ['PL_DEV_DEBUG'] = '1'
 
@@ -84,6 +84,7 @@ def test_model_checkpoint_period(tmpdir, epochs, period):
     )
     trainer.fit(model)
 
+    extra_on_train_end = (1 / period) % 1 > 0
     # check that the correct ckpts were created
-    expected_calls = epochs * int(1 / period)
+    expected_calls = epochs * int(1 / period) + int(extra_on_train_end)
     assert len(trainer.dev_debugger.checkpoint_callback_history) == expected_calls
\ No newline at end of file
diff --git a/tests/loggers/test_comet.py b/tests/loggers/test_comet.py
index 16e8d8551b6e5..f342628c8339f 100644
--- a/tests/loggers/test_comet.py
+++ b/tests/loggers/test_comet.py
@@ -96,7 +96,7 @@ def test_comet_logger_dirs_creation(tmpdir, monkeypatch):
     trainer.fit(model)
 
     assert trainer.checkpoint_callback.dirpath == (tmpdir / 'test' / version / 'checkpoints')
-    assert set(os.listdir(trainer.checkpoint_callback.dirpath)) == {'epoch=0.ckpt'}
+    assert set(os.listdir(trainer.checkpoint_callback.dirpath)) == {'epoch=0-step=9.ckpt'}
 
 
 def test_comet_name_default():
diff --git a/tests/loggers/test_mlflow.py b/tests/loggers/test_mlflow.py
index e5b871e4ec7be..8c6abca63ae82 100644
--- a/tests/loggers/test_mlflow.py
+++ b/tests/loggers/test_mlflow.py
@@ -41,7 +41,7 @@ def test_mlflow_logger_dirs_creation(tmpdir):
     assert 'epoch' in os.listdir(tmpdir / exp_id / run_id / 'metrics')
     assert set(os.listdir(tmpdir / exp_id / run_id / 'params')) == model.hparams.keys()
     assert trainer.checkpoint_callback.dirpath == (tmpdir / exp_id / run_id / 'checkpoints')
-    assert set(os.listdir(trainer.checkpoint_callback.dirpath)) == {'epoch=0.ckpt'}
+    assert set(os.listdir(trainer.checkpoint_callback.dirpath)) == {'epoch=0-step=9.ckpt'}
 
 
 def test_mlflow_experiment_id_retrieved_once(tmpdir):
diff --git a/tests/loggers/test_wandb.py b/tests/loggers/test_wandb.py
index 9907ad9d087a2..bccf4188dd291 100644
--- a/tests/loggers/test_wandb.py
+++ b/tests/loggers/test_wandb.py
@@ -95,4 +95,4 @@ def test_wandb_logger_dirs_creation(wandb, tmpdir):
     trainer.fit(model)
 
     assert trainer.checkpoint_callback.dirpath == str(tmpdir / 'project' / version / 'checkpoints')
-    assert set(os.listdir(trainer.checkpoint_callback.dirpath)) == {'epoch=0.ckpt'}
+    assert set(os.listdir(trainer.checkpoint_callback.dirpath)) == {'epoch=0-step=9.ckpt'}
diff --git a/tests/callbacks/test_model_checkpoint.py b/tests/models/test_checkpoint.py
similarity index 96%
rename from tests/callbacks/test_model_checkpoint.py
rename to tests/models/test_checkpoint.py
index 4445ba81127cf..eb101f9fdff23 100644
--- a/tests/callbacks/test_model_checkpoint.py
+++ b/tests/models/test_checkpoint.py
@@ -163,8 +163,8 @@ def test_model_checkpoint_format_checkpoint_name(tmpdir):
     ckpt_name = ModelCheckpoint(monitor='early_stop_on', filepath='').format_checkpoint_name(5, 4, {})
     assert ckpt_name == 'epoch=5-step=4.ckpt'
     # CWD
-    ckpt_name = ModelCheckpoint(monitor='early_stop_on', filepath='.').format_checkpoint_name(3, 4, {})
-    assert Path(ckpt_name) == Path('.') / 'epoch=3-step=4.ckpt'
+    ckpt_name = ModelCheckpoint(monitor='early_stop_on', filepath='../').format_checkpoint_name(3, 4, {})
+    assert Path(ckpt_name).absolute() == (Path('..') / 'epoch=3-step=4.ckpt').absolute()
     # dir does not exist so it is used as filename
     filepath = tmpdir / 'dir'
     ckpt_name = ModelCheckpoint(monitor='early_stop_on', filepath=filepath, prefix='test').format_checkpoint_name(3, 4, {})
@@ -183,14 +183,14 @@ def test_model_checkpoint_save_last(tmpdir):
     """Tests that save_last produces only one last checkpoint."""
     seed_everything()
     model = EvalModelTemplate()
-    epochs = 3
+    _chpt_name_last = ModelCheckpoint.CHECKPOINT_NAME_LAST
     ModelCheckpoint.CHECKPOINT_NAME_LAST = 'last-{epoch}'
     model_checkpoint = ModelCheckpoint(monitor='early_stop_on', filepath=tmpdir / '{step}', save_top_k=-1, save_last=True)
     trainer = Trainer(
         default_root_dir=tmpdir,
         early_stop_callback=False,
         checkpoint_callback=model_checkpoint,
-        max_epochs=epochs,
+        max_epochs=3,
         logger=False,
     )
     trainer.fit(model)
@@ -199,8 +199,8 @@ def test_model_checkpoint_save_last(tmpdir):
     )
     last_filename = last_filename + '.ckpt'
     assert str(tmpdir / last_filename) == model_checkpoint.last_model_path
-    assert set(os.listdir(tmpdir)) == set([f'step={i}.ckpt' for i in [19, 29, 30]] + [last_filename])
-    ModelCheckpoint.CHECKPOINT_NAME_LAST = 'last'
+    assert set(os.listdir(tmpdir)) == set([f'step={i}.ckpt' for i in [9, 19, 29]] + [last_filename])
+    ModelCheckpoint.CHECKPOINT_NAME_LAST = _chpt_name_last
 
 
 def test_invalid_top_k(tmpdir):
@@ -252,13 +252,13 @@ def test_model_checkpoint_none_monitor(tmpdir):
 
     # these should not be set if monitor is None
     assert checkpoint_callback.monitor is None
-    assert checkpoint_callback.best_model_path == checkpoint_callback.last_model_path == tmpdir / 'step=20.ckpt'
+    assert checkpoint_callback.best_model_path == checkpoint_callback.last_model_path == tmpdir / 'step=19.ckpt'
     assert checkpoint_callback.best_model_score == 0
     assert checkpoint_callback.best_k_models == {}
     assert checkpoint_callback.kth_best_model_path == ''
 
     # check that the correct ckpts were created
-    expected = [f'step={i}.ckpt' for i in [9, 19, 20]]
+    expected = [f'step={i}.ckpt' for i in [9, 19]]
     assert set(os.listdir(tmpdir)) == set(expected)
 
 
@@ -372,12 +372,12 @@ def test_default_checkpoint_behavior(tmpdir):
 
     assert len(results) == 1
     assert results[0]['test_acc'] >= 0.80
-    assert len(trainer.dev_debugger.checkpoint_callback_history) == 4
+    assert len(trainer.dev_debugger.checkpoint_callback_history) == 3
 
     # make sure the checkpoint we saved has the metric in the name
     ckpts = os.listdir(os.path.join(tmpdir, 'lightning_logs', 'version_0', 'checkpoints'))
     assert len(ckpts) == 1
-    assert ckpts[0] == 'epoch=2-step=15.ckpt'
+    assert ckpts[0] == 'epoch=2-step=14.ckpt'
 
 
 def test_ckpt_metric_names_results(tmpdir):
@@ -448,9 +448,10 @@ def test_model_checkpoint_save_last_checkpoint_contents(tmpdir):
     path_last_epoch = str(tmpdir / f"epoch={num_epochs - 1}.ckpt")
     path_last = str(tmpdir / "last.ckpt")
     assert path_last == model_checkpoint.last_model_path
-    assert os.path.isfile(path_last_epoch)
 
+    assert os.path.isfile(path_last_epoch)
     ckpt_last_epoch = torch.load(path_last_epoch)
+    assert os.path.isfile(path_last)
     ckpt_last = torch.load(path_last)
     assert all(ckpt_last_epoch[k] == ckpt_last[k] for k in ("epoch", "global_step"))
 
@@ -532,7 +533,12 @@ def mock_save_function(filepath, *args):
     losses = [10, 9, 2.8, 5, 2.5]
 
     checkpoint_callback = ModelCheckpoint(
-        tmpdir, monitor='checkpoint_on', save_top_k=save_top_k, save_last=save_last, prefix=file_prefix, verbose=1
+        tmpdir / '{epoch}',
+        monitor='checkpoint_on',
+        save_top_k=save_top_k,
+        save_last=save_last,
+        prefix=file_prefix,
+        verbose=1,
     )
     checkpoint_callback.save_function = mock_save_function
     trainer = Trainer()
diff --git a/tests/trainer/deprecate_legacy_flow/__init__.py b/tests/trainer/deprecate_legacy_flow/__init__.py
new file mode 100644
index 0000000000000..e69de29bb2d1d