diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index f7c87ec1..73d33e17 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -37,3 +37,25 @@ repos:
       # Run the formatter
       - id: ruff-format
         types_or: [python, pyi]
+
+  - repo: https://github.com/pre-commit/mirrors-mypy
+    rev: "v1.10.0"
+    hooks:
+      - id: mypy
+        files: plum
+        additional_dependencies:
+          - pytest
+        exclude: |
+          (?x)
+          ^(
+              | plum/alias\.py
+              | plum/function\.py
+              | plum/method\.py
+              | plum/parametric\.py
+              | plum/promotion\.py
+              | plum/repr\.py
+              | plum/resolver\.py
+              | plum/signature\.py
+              | plum/type\.py
+              | plum/util\.py
+          )$
diff --git a/plum/__init__.py b/plum/__init__.py
index 543a620a..2d9bc308 100644
--- a/plum/__init__.py
+++ b/plum/__init__.py
@@ -2,7 +2,9 @@
 # versions from `typing`. To not break backward compatibility, we still export these
 # types.
 from functools import partial
-from typing import Dict, List, Tuple, Union  # noqa: F401
+from typing import Dict, List, Tuple, Type, TypeVar, Union  # noqa: F401
+
+from typing_extensions import TypeGuard  # noqa: F401
 
 from beartype import BeartypeConf as _BeartypeConf
 from beartype import BeartypeStrategy as _BeartypeStrategy
@@ -31,8 +33,14 @@
 # actually is not yet available, but we can already opt in to use it.
 _is_bearable = partial(_is_bearable, conf=_BeartypeConf(strategy=_BeartypeStrategy.On))
 
+T = TypeVar("T")
+T2 = TypeVar("T2")
+
 
-def isinstance(instance, c):
+# TODO: TypeGuard or
+def isinstance(
+    instance: object, c: Union[Type[T], _TypeHint[T]]
+) -> TypeGuard[Union[T, _TypeHint[T]]]:
     """Check if `instance` is of type or type hint `c`.
 
     This is a drop-in replace for the built-in :func:`ininstance` which supports type
@@ -45,10 +53,13 @@ def isinstance(instance, c):
     Returns:
         bool: Whether `instance` is of type or type hint `c`.
     """
-    return _is_bearable(instance, resolve_type_hint(c))
+    pred: bool = _is_bearable(instance, resolve_type_hint(c))
+    return pred
 
 
-def issubclass(c1, c2):
+def issubclass(
+    c1: Union[Type[T], _TypeHint[T]], c2: Union[Type[T2], _TypeHint[T2]]
+) -> TypeGuard[Union[T2, _TypeHint[T2]]]:
     """Check if `c1` is a subclass or sub-type hint of `c2`.
 
     This is a drop-in replace for the built-in :func:`issubclass` which supports type
@@ -61,4 +72,5 @@ def issubclass(c1, c2):
     Returns:
         bool: Whether `c1` is a subtype or sub-type hint of `c2`.
     """
-    return _TypeHint(resolve_type_hint(c1)) <= _TypeHint(resolve_type_hint(c2))
+    pred: bool = _TypeHint(resolve_type_hint(c1)) <= _TypeHint(resolve_type_hint(c2))
+    return pred
diff --git a/plum/autoreload.py b/plum/autoreload.py
index 21d20e7d..f17f7278 100644
--- a/plum/autoreload.py
+++ b/plum/autoreload.py
@@ -6,7 +6,7 @@
 __all__ = ["activate_autoreload", "deactivate_autoreload"]
 
 
-def _update_instances(old, new):
+def _update_instances(old: type, new: type) -> None:
     """First call the original implementation of Autoreload's :meth:`update_instances`,
     and then use :obj:`.type._type_mapping` to map type `old` to the type `new`.
 
@@ -14,7 +14,7 @@ def _update_instances(old, new):
         old (type): Old type.
         new (type): New type.
     """
-    _update_instances_original(old, new)
+    _update_instances_original(old, new)  # type: ignore[misc]
 
     type_mapping[old] = new
 
@@ -33,10 +33,10 @@ def _update_instances(old, new):
 """function: Original implementation of :func:`update_instances`."""
 
 
-def activate_autoreload():
+def activate_autoreload() -> None:
     """Pirate Autoreload's `update_instance` function to have Plum work with
     Autoreload."""
-    from IPython.extensions import autoreload  # type: ignore
+    from IPython.extensions import autoreload
 
     # First, cache the original method so we can deactivate ourselves.
     global _update_instances_original
@@ -47,7 +47,7 @@ def activate_autoreload():
     autoreload.update_instances = _update_instances
 
 
-def deactivate_autoreload():
+def deactivate_autoreload() -> None:
     """Disable Plum's autoreload hack. This undoes what
     :func:`.autoreload.activate_autoreload` did."""
     global _update_instances_original
@@ -66,7 +66,7 @@ def deactivate_autoreload():
     try:
         # Try to load iPython and get the iPython session, but don't crash if
         # this does not work (for example IPython not installed, or python shell)
-        from IPython import get_ipython  # type: ignore
+        from IPython import get_ipython
 
         ip = get_ipython()
         ext_loaded = "IPython.extensions.storemagic" in ip.extension_manager.loaded
diff --git a/plum/dispatcher.py b/plum/dispatcher.py
index f81d650d..c69c5c2e 100644
--- a/plum/dispatcher.py
+++ b/plum/dispatcher.py
@@ -32,10 +32,10 @@ class Dispatcher:
     classes: Dict[str, Dict[str, Function]] = field(default_factory=dict)
 
     @overload
-    def __call__(self, method: T, precedence: int = ...) -> T: ...
+    def __call__(self, method: None, precedence: int) -> Callable[[T], T]: ...
 
     @overload
-    def __call__(self, method: None, precedence: int) -> Callable[[T], T]: ...
+    def __call__(self, method: T, precedence: int = ...) -> T: ...
 
     def __call__(
         self, method: Optional[T] = None, precedence: int = 0
@@ -130,13 +130,13 @@ def _add_method(
             f.register(method, signature, precedence)
         return f
 
-    def clear_cache(self):
+    def clear_cache(self) -> None:
         """Clear cache."""
         for f in self.functions.values():
             f.clear_cache()
 
 
-def clear_all_cache():
+def clear_all_cache() -> None:
     """Clear all cache, including the cache of subclass checks. This should be called
     if types are modified."""
     for f in Function._instances:
diff --git a/pyproject.toml b/pyproject.toml
index 5c04c32a..d7f8b140 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -62,6 +62,15 @@ branch = true
 command_line = "-m pytest --verbose test"
 source = ["plum"]
 
+
+[tool.mypy]
+  disable_error_code = ["no-redef"]
+  enable_error_code = ["ignore-without-code", "redundant-expr", "truthy-bool"]
+  files = ["plum"]
+  strict = true
+  follow_imports = "skip"
+
+
 [tool.pytest.ini_options]
 testpaths = ["tests/", "plum", "docs"]
 addopts = [
diff --git a/tests/typechecked/test_overload.py b/tests/typechecked/test_overload.py
index 2bbb5135..074f846d 100644
--- a/tests/typechecked/test_overload.py
+++ b/tests/typechecked/test_overload.py
@@ -23,7 +23,7 @@ def f(x):  # E: pyright(overloaded implementation is not consistent)
 def test_overload() -> None:
     assert f(1) == 1
     assert f("1") == "1"
-    with pytest.raises(NotFoundLookupError):
+    with pytest.raises(NotFoundLookupError, match="no overload variant"):
         # E: pyright(argument of type "float" cannot be assigned to parameter "x")
         # E: pyright(no overloads for "f" match the provided arguments)
         # E: mypy(no overload variant of "f" matches argument type "float")