diff --git a/ignite/engine/engine.py b/ignite/engine/engine.py
index ab9e137b59f8..3bf935eea6d3 100644
--- a/ignite/engine/engine.py
+++ b/ignite/engine/engine.py
@@ -438,14 +438,28 @@ def fire_event(self, event_name: Any) -> None:
         return self._fire_event(event_name)
 
     def terminate(self) -> None:
-        """Sends terminate signal to the engine, so that it terminates completely the run after
-        the current iteration."""
+        """Sends terminate signal to the engine, so that it terminates completely the run. The run is
+        terminated after the event on which ``terminate`` method was called. The following events are triggered:
+
+        - ...
+        - Terminating event
+        - :attr:`~ignite.engine.events.Events.TERMINATE`
+        - :attr:`~ignite.engine.events.Events.COMPLETED`
+        """
         self.logger.info("Terminate signaled. Engine will stop after current iteration is finished.")
         self.should_terminate = True
 
     def terminate_epoch(self) -> None:
-        """Sends terminate signal to the engine, so that it terminates the current epoch
-        after the current iteration."""
+        """Sends terminate signal to the engine, so that it terminates the current epoch. The run
+        continues from the next epoch. The following events are triggered:
+
+        - ...
+        - Event on which ``terminate_epoch`` method is called
+        - :attr:`~ignite.engine.events.Events.TERMINATE_SINGLE_EPOCH`
+        - :attr:`~ignite.engine.events.Events.EPOCH_COMPLETED`
+        - :attr:`~ignite.engine.events.Events.EPOCH_STARTED`
+        - ...
+        """
         self.logger.info(
             "Terminate current epoch is signaled. "
             "Current epoch iteration will stop after current iteration is finished."
diff --git a/tests/ignite/engine/test_engine.py b/tests/ignite/engine/test_engine.py
index b038a3ce7cae..8d7d21f686f2 100644
--- a/tests/ignite/engine/test_engine.py
+++ b/tests/ignite/engine/test_engine.py
@@ -104,7 +104,7 @@ def end_of_epoch_handler(engine):
 
 
 @pytest.mark.parametrize("data, epoch_length", [(None, 10), (range(10), None)])
-def test_terminate_at_start_of_epoch_stops_run_after_completing_iteration(data, epoch_length):
+def test_terminate_at_start_of_epoch(data, epoch_length):
     max_epochs = 5
     epoch_to_terminate_on = 3
     real_epoch_length = epoch_length if data is None else len(data)
@@ -293,7 +293,7 @@ def start_of_iteration_handler(engine):
 def test_terminate_epoch_events_sequence(terminate_epoch_event, i):
     engine = RecordedEngine(MagicMock(return_value=1))
     data = range(10)
-    max_epochs = 5
+    max_epochs = 3
 
     # TODO: Bug: Events.GET_BATCH_STARTED(once=12) is called twice !
     # prevent call_terminate_epoch to be called twice
@@ -310,14 +310,23 @@ def call_terminate_epoch():
     def check_previous_events(iter_counter):
         e = i // len(data) + 1
 
-        print("engine.called_events:", engine.called_events)
-
         assert engine.called_events[0] == (0, 0, Events.STARTED)
-        assert engine.called_events[-1] == (e, i, Events.TERMINATE_SINGLE_EPOCH)
         assert engine.called_events[-2] == (e, i, terminate_epoch_event)
+        assert engine.called_events[-1] == (e, i, Events.TERMINATE_SINGLE_EPOCH)
+
+    @engine.on(Events.EPOCH_COMPLETED)
+    def check_previous_events2():
+        e = i // len(data) + 1
+        if e == engine.state.epoch and i == engine.state.iteration:
+            assert engine.called_events[-3] == (e, i, terminate_epoch_event)
+            assert engine.called_events[-2] == (e, i, Events.TERMINATE_SINGLE_EPOCH)
+            assert engine.called_events[-1] == (e, i, Events.EPOCH_COMPLETED)
 
     engine.run(data, max_epochs=max_epochs)
 
+    assert engine.state.epoch == max_epochs
+    assert (max_epochs - 1) * len(data) < engine.state.iteration < max_epochs * len(data)
+
 
 def _create_mock_data_loader(epochs, batches_per_epoch):
     batches = [MagicMock()] * batches_per_epoch