diff --git a/invoke/collection.py b/invoke/collection.py
index 23dcff928..20ef3576c 100644
--- a/invoke/collection.py
+++ b/invoke/collection.py
@@ -1,6 +1,6 @@
 import copy
 from types import ModuleType
-from typing import Any, Callable, Dict, List, Optional, Tuple
+from typing import Any, Callable, Dict, List, Optional, Tuple, Union
 
 from .util import Lexicon, helpline
 
@@ -284,7 +284,7 @@ def add_task(
 
     def add_collection(
         self,
-        coll: "Collection",
+        coll: Union["Collection", ModuleType],
         name: Optional[str] = None,
         default: Optional[bool] = None,
     ) -> None:
diff --git a/invoke/context.py b/invoke/context.py
index e9beaf4d1..21c5101e3 100644
--- a/invoke/context.py
+++ b/invoke/context.py
@@ -389,7 +389,7 @@ def cd(self, path: Union[PathLike, str]) -> Generator[None, None, None]:
             (such as the various ``Path`` objects out there), and not just
             string literals.
         """
-        path = str(path)
+        path = os.fspath(path)
         self.command_cwds.append(path)
         try:
             yield
diff --git a/invoke/parser/context.py b/invoke/parser/context.py
index 359e9f9e2..1d44712c5 100644
--- a/invoke/parser/context.py
+++ b/invoke/parser/context.py
@@ -91,7 +91,7 @@ def __init__(
         self.args = Lexicon()
         self.positional_args: List[Argument] = []
         self.flags = Lexicon()
-        self.inverse_flags: Dict[str, str] = {}  # No need for Lexicon here
+        self.inverse_flags: Dict[str, Any] = {}  # No need for Lexicon here
         self.name = name
         self.aliases = aliases
         for arg in args:
diff --git a/invoke/runners.py b/invoke/runners.py
index f1c888f44..5ac3cd52f 100644
--- a/invoke/runners.py
+++ b/invoke/runners.py
@@ -19,6 +19,7 @@
     Optional,
     Tuple,
     Type,
+    overload,
 )
 
 # Import some platform-specific things at top level so they can be mocked for
@@ -122,6 +123,16 @@ def __init__(self, context: "Context") -> None:
         self._asynchronous = False
         self._disowned = False
 
+    @overload
+    def run(self, command: str, *, disowned: None, **kwargs: Any) -> "Result":
+        ...
+
+    @overload
+    def run(
+        self, command: str, *, disowned: bool, **kwargs: Any
+    ) -> Optional["Result"]:
+        ...
+
     def run(self, command: str, **kwargs: Any) -> Optional["Result"]:
         """
         Execute ``command``, returning an instance of `Result` once complete.
@@ -1481,7 +1492,7 @@ def __init__(
         exited: int = 0,
         pty: bool = False,
         hide: Tuple[str, ...] = tuple(),
-    ):
+    ) -> None:
         self.stdout = stdout
         self.stderr = stderr
         if encoding is None:
@@ -1506,6 +1517,9 @@ def return_code(self) -> int:
     def __bool__(self) -> bool:
         return self.ok
 
+    def __int__(self) -> int:
+        return self.exited
+
     def __str__(self) -> str:
         if self.exited is not None:
             desc = "Command exited with status {}.".format(self.exited)
diff --git a/invoke/tasks.py b/invoke/tasks.py
index cd3075e9b..a967a2836 100644
--- a/invoke/tasks.py
+++ b/invoke/tasks.py
@@ -21,6 +21,7 @@
     Type,
     TypeVar,
     Union,
+    cast,
 )
 
 from .context import Context
@@ -286,7 +287,7 @@ def get_arguments(
         return args
 
 
-def task(*args: Any, **kwargs: Any) -> Callable:
+def task(*args: Any, **kwargs: Any) -> "Task[T]":
     """
     Marks wrapped callable object as a valid Invoke task.
 
@@ -357,7 +358,7 @@ def inner(body: Callable) -> Task[T]:
         return _task
 
     # update_wrapper(inner, klass)
-    return inner
+    return cast('Task["T"]', inner)
 
 
 class Call:
diff --git a/tasks.py b/tasks.py
index 12e0c3b84..ce8459971 100644
--- a/tasks.py
+++ b/tasks.py
@@ -53,7 +53,7 @@ def test(
 
 # TODO: replace with invocations' once the "call truly local tester" problem is
 # solved (see other TODOs). For now this is just a copy/paste/modify.
-@task(help=test.help)  # type: ignore
+@task(help=test.help)
 def integration(
     c: "Context", opts: Optional[str] = None, pty: bool = True
 ) -> None:
diff --git a/tests/context.py b/tests/context.py
index b7266042a..d1b031881 100644
--- a/tests/context.py
+++ b/tests/context.py
@@ -213,7 +213,7 @@ class Path:
                 def __init__(self, value):
                     self.value = value
 
-                def __str__(self):
+                def __fspath__(self):
                     return self.value
 
             runner = Local.return_value