diff --git a/.travis.yml b/.travis.yml
index 89102f4a..1588ee1b 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -30,8 +30,8 @@ matrix:
       install:
         - 'pip install .[qa]'
       script:
-        # Ignore F401, which are unused imports. flake8 is a primitive tool and is sometimes wrong.
-        - 'flake8 --extend-ignore F401 jedi setup.py'
+        - 'flake8 jedi setup.py'
+        - 'mypy jedi sith.py'
 install:
     - sudo apt-get -y install python3-venv
     - pip install .[testing]
diff --git a/jedi/api/classes.py b/jedi/api/classes.py
index 05ee3885..a8d9fb19 100644
--- a/jedi/api/classes.py
+++ b/jedi/api/classes.py
@@ -15,18 +15,18 @@ the interesting information about all operations.
 """
 import re
 import warnings
+from pathlib import Path
 from typing import Optional
 
-from parso.python.tree import search_ancestor
+from parso.tree import search_ancestor
 
 from jedi import settings
 from jedi import debug
 from jedi.inference.utils import unite
 from jedi.cache import memoize_method
-from jedi.inference import imports
-from jedi.inference.imports import ImportName
 from jedi.inference.compiled.mixed import MixedName
-from jedi.inference.gradual.typeshed import StubModuleValue
+from jedi.inference.names import ImportName, SubModuleName
+from jedi.inference.gradual.stub_value import StubModuleValue
 from jedi.inference.gradual.conversion import convert_names, convert_values
 from jedi.inference.base_value import ValueSet
 from jedi.api.keywords import KeywordName
@@ -93,17 +93,15 @@ class BaseName:
         return self._name.get_root_context()
 
     @property
-    def module_path(self) -> Optional[str]:
+    def module_path(self) -> Optional[Path]:
         """
         Shows the file path of a module. e.g. ``/usr/lib/python3.9/os.py``
-
-        :rtype: str or None
         """
         module = self._get_module_context()
         if module.is_stub() or not module.is_compiled():
             # Compiled modules should not return a module path even if they
             # have one.
-            path = self._get_module_context().py__file__()
+            path: Optional[Path] = self._get_module_context().py__file__()
             if path is not None:
                 return path
 
@@ -186,7 +184,7 @@ class BaseName:
                     tree_name.is_definition():
                 resolve = True
 
-        if isinstance(self._name, imports.SubModuleName) or resolve:
+        if isinstance(self._name, SubModuleName) or resolve:
             for value in self._name.infer():
                 return value.api_type
         return self._name.api_type
@@ -497,7 +495,7 @@ class BaseName:
         return [self if n == self._name else Name(self._inference_state, n)
                 for n in resulting_names]
 
-    @property
+    @property  # type: ignore[misc]
     @memoize_method
     def params(self):
         warnings.warn(
diff --git a/jedi/api/completion_cache.py b/jedi/api/completion_cache.py
index 6dac6a4e..46e9bead 100644
--- a/jedi/api/completion_cache.py
+++ b/jedi/api/completion_cache.py
@@ -1,7 +1,13 @@
-_cache = {}
+from typing import Dict, Tuple, Callable
+
+CacheValues = Tuple[str, str, str]
+CacheValuesCallback = Callable[[], CacheValues]
 
 
-def save_entry(module_name, name, cache):
+_cache: Dict[str, Dict[str, CacheValues]] = {}
+
+
+def save_entry(module_name: str, name: str, cache: CacheValues) -> None:
     try:
         module_cache = _cache[module_name]
     except KeyError:
@@ -9,8 +15,8 @@ def save_entry(module_name, name, cache):
     module_cache[name] = cache
 
 
-def _create_get_from_cache(number):
-    def _get_from_cache(module_name, name, get_cache_values):
+def _create_get_from_cache(number: int) -> Callable[[str, str, CacheValuesCallback], str]:
+    def _get_from_cache(module_name: str, name: str, get_cache_values: CacheValuesCallback) -> str:
         try:
             return _cache[module_name][name][number]
         except KeyError:
diff --git a/jedi/api/environment.py b/jedi/api/environment.py
index f7636140..3f1f238f 100644
--- a/jedi/api/environment.py
+++ b/jedi/api/environment.py
@@ -384,7 +384,8 @@ def _get_executable_path(path, safe=True):
 
 
 def _get_executables_from_windows_registry(version):
-    import winreg
+    # https://github.com/python/typeshed/pull/3794 adds winreg
+    import winreg  # type: ignore[import]
 
     # TODO: support Python Anaconda.
     sub_keys = [
diff --git a/jedi/api/keywords.py b/jedi/api/keywords.py
index a0146de0..95c49227 100644
--- a/jedi/api/keywords.py
+++ b/jedi/api/keywords.py
@@ -1,10 +1,13 @@
 import pydoc
 from contextlib import suppress
+from typing import Dict, Optional
 
 from jedi.inference.names import AbstractArbitraryName
 
 try:
-    from pydoc_data import topics as pydoc_topics
+    # https://github.com/python/typeshed/pull/4351 adds pydoc_data
+    from pydoc_data import topics  # type: ignore[import]
+    pydoc_topics: Optional[Dict[str, str]] = topics.topics
 except ImportError:
     # Python 3.6.8 embeddable does not have pydoc_data.
     pydoc_topics = None
@@ -44,6 +47,6 @@ def imitate_pydoc(string):
         return ''
 
     try:
-        return pydoc_topics.topics[label].strip() if pydoc_topics else ''
+        return pydoc_topics[label].strip() if pydoc_topics else ''
     except KeyError:
         return ''
diff --git a/jedi/cache.py b/jedi/cache.py
index 29ed979f..1ff45201 100644
--- a/jedi/cache.py
+++ b/jedi/cache.py
@@ -13,14 +13,15 @@ these variables are being cleaned after every API usage.
 """
 import time
 from functools import wraps
+from typing import Any, Dict, Tuple
 
 from jedi import settings
 from parso.cache import parser_cache
 
-_time_caches = {}
+_time_caches: Dict[str, Dict[Any, Tuple[float, Any]]] = {}
 
 
-def clear_time_caches(delete_all=False):
+def clear_time_caches(delete_all: bool = False) -> None:
     """ Jedi caches many things, that should be completed after each completion
     finishes.
 
diff --git a/jedi/debug.py b/jedi/debug.py
index 4baa0d6a..2b2bcfb8 100644
--- a/jedi/debug.py
+++ b/jedi/debug.py
@@ -1,6 +1,7 @@
 import os
 import time
 from contextlib import contextmanager
+from typing import Callable, Optional
 
 _inited = False
 
@@ -20,7 +21,7 @@ try:
         raise ImportError
     else:
         # Use colorama for nicer console output.
-        from colorama import Fore, init
+        from colorama import Fore, init  # type: ignore[import]
         from colorama import initialise
 
         def _lazy_colorama_init():  # noqa: F811
@@ -45,7 +46,7 @@ try:
             _inited = True
 
 except ImportError:
-    class Fore:
+    class Fore:  # type: ignore[no-redef]
         RED = ''
         GREEN = ''
         YELLOW = ''
@@ -62,7 +63,7 @@ enable_warning = False
 enable_notice = False
 
 # callback, interface: level, str
-debug_function = None
+debug_function: Optional[Callable[[str, str], None]] = None
 _debug_indent = 0
 _start_time = time.time()
 
diff --git a/jedi/inference/__init__.py b/jedi/inference/__init__.py
index 3461f502..8cf6a1c2 100644
--- a/jedi/inference/__init__.py
+++ b/jedi/inference/__init__.py
@@ -120,14 +120,15 @@ class InferenceState:
         debug.dbg('execute result: %s in %s', value_set, value)
         return value_set
 
-    @property
+    # mypy doesn't suppport decorated propeties (https://github.com/python/mypy/issues/1362)
+    @property  # type: ignore[misc]
     @inference_state_function_cache()
     def builtins_module(self):
         module_name = 'builtins'
         builtins_module, = self.import_module((module_name,), sys_path=())
         return builtins_module
 
-    @property
+    @property  # type: ignore[misc]
     @inference_state_function_cache()
     def typing_module(self):
         typing_module, = self.import_module(('typing',))
diff --git a/jedi/inference/compiled/__init__.py b/jedi/inference/compiled/__init__.py
index faf5d373..09ac19f9 100644
--- a/jedi/inference/compiled/__init__.py
+++ b/jedi/inference/compiled/__init__.py
@@ -1,3 +1,6 @@
+# This file also re-exports symbols for wider use. We configure mypy and flake8
+# to be aware that this file does this.
+
 from jedi.inference.compiled.value import CompiledValue, CompiledName, \
     CompiledValueFilter, CompiledValueName, create_from_access_path
 from jedi.inference.base_value import LazyValueWrapper
diff --git a/jedi/inference/compiled/subprocess/__init__.py b/jedi/inference/compiled/subprocess/__init__.py
index 13366278..0795de30 100644
--- a/jedi/inference/compiled/subprocess/__init__.py
+++ b/jedi/inference/compiled/subprocess/__init__.py
@@ -29,19 +29,19 @@ _MAIN_PATH = os.path.join(os.path.dirname(__file__), '__main__.py')
 PICKLE_PROTOCOL = 4
 
 
-class _GeneralizedPopen(subprocess.Popen):
-    def __init__(self, *args, **kwargs):
-        if os.name == 'nt':
-            try:
-                # Was introduced in Python 3.7.
-                CREATE_NO_WINDOW = subprocess.CREATE_NO_WINDOW
-            except AttributeError:
-                CREATE_NO_WINDOW = 0x08000000
-            kwargs['creationflags'] = CREATE_NO_WINDOW
-        # The child process doesn't need file descriptors except 0, 1, 2.
-        # This is unix only.
-        kwargs['close_fds'] = 'posix' in sys.builtin_module_names
-        super().__init__(*args, **kwargs)
+def _GeneralizedPopen(*args, **kwargs):
+    if os.name == 'nt':
+        try:
+            # Was introduced in Python 3.7.
+            CREATE_NO_WINDOW = subprocess.CREATE_NO_WINDOW
+        except AttributeError:
+            CREATE_NO_WINDOW = 0x08000000
+        kwargs['creationflags'] = CREATE_NO_WINDOW
+    # The child process doesn't need file descriptors except 0, 1, 2.
+    # This is unix only.
+    kwargs['close_fds'] = 'posix' in sys.builtin_module_names
+
+    return subprocess.Popen(*args, **kwargs)
 
 
 def _enqueue_output(out, queue_):
diff --git a/jedi/inference/compiled/subprocess/__main__.py b/jedi/inference/compiled/subprocess/__main__.py
index 370e2361..e1554334 100644
--- a/jedi/inference/compiled/subprocess/__main__.py
+++ b/jedi/inference/compiled/subprocess/__main__.py
@@ -1,5 +1,6 @@
 import os
 import sys
+from importlib.abc import MetaPathFinder
 from importlib.machinery import PathFinder
 
 # Remove the first entry, because it's simply a directory entry that equals
@@ -16,7 +17,7 @@ def _get_paths():
     return {'jedi': _jedi_path, 'parso': _parso_path}
 
 
-class _ExactImporter:
+class _ExactImporter(MetaPathFinder):
     def __init__(self, path_dct):
         self._path_dct = path_dct
 
diff --git a/jedi/inference/docstrings.py b/jedi/inference/docstrings.py
index e9c94cdc..ee7a8d89 100644
--- a/jedi/inference/docstrings.py
+++ b/jedi/inference/docstrings.py
@@ -50,7 +50,7 @@ def _get_numpy_doc_string_cls():
     global _numpy_doc_string_cache
     if isinstance(_numpy_doc_string_cache, (ImportError, SyntaxError)):
         raise _numpy_doc_string_cache
-    from numpydoc.docscrape import NumpyDocString
+    from numpydoc.docscrape import NumpyDocString  # type: ignore[import]
     _numpy_doc_string_cache = NumpyDocString
     return _numpy_doc_string_cache
 
diff --git a/jedi/inference/filters.py b/jedi/inference/filters.py
index a796c445..2451f468 100644
--- a/jedi/inference/filters.py
+++ b/jedi/inference/filters.py
@@ -3,9 +3,11 @@ Filters are objects that you can use to filter names in different scopes. They
 are needed for name resolution.
 """
 from abc import abstractmethod
+from typing import List, MutableMapping, Type
 import weakref
 
 from parso.tree import search_ancestor
+from parso.python.tree import Name, UsedNamesMapping
 
 from jedi.inference import flow_analysis
 from jedi.inference.base_value import ValueSet, ValueWrapper, \
@@ -13,8 +15,9 @@ from jedi.inference.base_value import ValueSet, ValueWrapper, \
 from jedi.parser_utils import get_cached_parent_scope
 from jedi.inference.utils import to_list
 from jedi.inference.names import TreeNameDefinition, ParamName, \
-    AnonymousParamName, AbstractNameDefinition
+    AnonymousParamName, AbstractNameDefinition, NameWrapper
 
+_definition_name_cache: MutableMapping[UsedNamesMapping, List[Name]]
 _definition_name_cache = weakref.WeakKeyDictionary()
 
 
@@ -36,7 +39,7 @@ class AbstractFilter:
 
 
 class FilterWrapper:
-    name_wrapper_class = None
+    name_wrapper_class: Type[NameWrapper]
 
     def __init__(self, wrapped_filter):
         self._wrapped_filter = wrapped_filter
diff --git a/jedi/inference/flow_analysis.py b/jedi/inference/flow_analysis.py
index d5369711..89bfe578 100644
--- a/jedi/inference/flow_analysis.py
+++ b/jedi/inference/flow_analysis.py
@@ -1,12 +1,14 @@
+from typing import Dict, Optional
+
 from jedi.parser_utils import get_flow_branch_keyword, is_scope, get_parent_scope
 from jedi.inference.recursion import execution_allowed
 from jedi.inference.helpers import is_big_annoying_library
 
 
 class Status:
-    lookup_table = {}
+    lookup_table: Dict[Optional[bool], 'Status'] = {}
 
-    def __init__(self, value, name):
+    def __init__(self, value: Optional[bool], name: str) -> None:
         self._value = value
         self._name = name
         Status.lookup_table[value] = self
diff --git a/jedi/inference/gradual/typeshed.py b/jedi/inference/gradual/typeshed.py
index e186bfc3..6295294d 100644
--- a/jedi/inference/gradual/typeshed.py
+++ b/jedi/inference/gradual/typeshed.py
@@ -2,6 +2,7 @@ import os
 import re
 from functools import wraps
 from collections import namedtuple
+from typing import Dict, Mapping, Tuple
 from pathlib import Path
 
 from jedi import settings
@@ -74,7 +75,7 @@ def _get_typeshed_directories(version_info):
             yield PathInfo(str(base_path.joinpath(check_version)), is_third_party)
 
 
-_version_cache = {}
+_version_cache: Dict[Tuple[int, int], Mapping[str, PathInfo]] = {}
 
 
 def _cache_stub_file_map(version_info):
diff --git a/jedi/inference/gradual/utils.py b/jedi/inference/gradual/utils.py
index 83162625..1fc90e7f 100644
--- a/jedi/inference/gradual/utils.py
+++ b/jedi/inference/gradual/utils.py
@@ -1,4 +1,3 @@
-import os
 from pathlib import Path
 
 from jedi.inference.gradual.typeshed import TYPESHED_PATH, create_stub_module
diff --git a/jedi/inference/names.py b/jedi/inference/names.py
index 02d75ea4..38be6f22 100644
--- a/jedi/inference/names.py
+++ b/jedi/inference/names.py
@@ -1,5 +1,6 @@
 from abc import abstractmethod
 from inspect import Parameter
+from typing import Optional, Tuple
 
 from parso.tree import search_ancestor
 
@@ -24,8 +25,8 @@ def _merge_name_docs(names):
 
 
 class AbstractNameDefinition:
-    start_pos = None
-    string_name = None
+    start_pos: Optional[Tuple[int, int]] = None
+    string_name: str
     parent_context = None
     tree_name = None
     is_value_name = True
diff --git a/jedi/inference/references.py b/jedi/inference/references.py
index 07a840a1..7c77f6f8 100644
--- a/jedi/inference/references.py
+++ b/jedi/inference/references.py
@@ -5,7 +5,8 @@ from parso import python_bytes_to_unicode
 
 from jedi.debug import dbg
 from jedi.file_io import KnownContentFileIO
-from jedi.inference.imports import SubModuleName, load_module_from_path
+from jedi.inference.names import SubModuleName
+from jedi.inference.imports import load_module_from_path
 from jedi.inference.filters import ParserTreeFilter
 from jedi.inference.gradual.conversion import convert_names
 
diff --git a/jedi/inference/sys_path.py b/jedi/inference/sys_path.py
index 0f44be4a..48b9ac89 100644
--- a/jedi/inference/sys_path.py
+++ b/jedi/inference/sys_path.py
@@ -14,8 +14,8 @@ from jedi import debug
 _BUILDOUT_PATH_INSERTION_LIMIT = 10
 
 
-def _abs_path(module_context, path: str):
-    path = Path(path)
+def _abs_path(module_context, str_path: str):
+    path = Path(str_path)
     if path.is_absolute():
         return path
 
diff --git a/jedi/inference/value/__init__.py b/jedi/inference/value/__init__.py
index 2e17ba26..62164215 100644
--- a/jedi/inference/value/__init__.py
+++ b/jedi/inference/value/__init__.py
@@ -1,3 +1,6 @@
+# Re-export symbols for wider use. We configure mypy and flake8 to be aware that
+# this file does this.
+
 from jedi.inference.value.module import ModuleValue
 from jedi.inference.value.klass import ClassValue
 from jedi.inference.value.function import FunctionValue, \
diff --git a/jedi/inference/value/instance.py b/jedi/inference/value/instance.py
index 153ab4f4..9514e01d 100644
--- a/jedi/inference/value/instance.py
+++ b/jedi/inference/value/instance.py
@@ -1,6 +1,6 @@
 from abc import abstractproperty
 
-from parso.python.tree import search_ancestor
+from parso.tree import search_ancestor
 
 from jedi import debug
 from jedi import settings
diff --git a/jedi/inference/value/module.py b/jedi/inference/value/module.py
index 128f7185..a5073b0e 100644
--- a/jedi/inference/value/module.py
+++ b/jedi/inference/value/module.py
@@ -1,5 +1,6 @@
 import os
 from pathlib import Path
+from typing import Optional
 
 from jedi.inference.cache import inference_state_method_cache
 from jedi.inference.names import AbstractNameDefinition, ModuleName
@@ -79,7 +80,7 @@ class ModuleMixin(SubModuleDictMixin):
     def is_stub(self):
         return False
 
-    @property
+    @property  # type: ignore[misc]
     @inference_state_method_cache()
     def name(self):
         return self._module_name_class(self, self.string_names[-1])
@@ -145,7 +146,7 @@ class ModuleValue(ModuleMixin, TreeValue):
         )
         self.file_io = file_io
         if file_io is None:
-            self._path = None
+            self._path: Optional[Path] = None
         else:
             self._path = Path(file_io.path)
         self.string_names = string_names  # Optional[Tuple[str, ...]]
@@ -165,7 +166,7 @@ class ModuleValue(ModuleMixin, TreeValue):
             return None
         return '.'.join(self.string_names)
 
-    def py__file__(self) -> Path:
+    def py__file__(self) -> Optional[Path]:
         """
         In contrast to Python's __file__ can be None.
         """
diff --git a/jedi/inference/value/namespace.py b/jedi/inference/value/namespace.py
index 0ea9ecf4..fe629a07 100644
--- a/jedi/inference/value/namespace.py
+++ b/jedi/inference/value/namespace.py
@@ -38,7 +38,7 @@ class ImplicitNamespaceValue(Value, SubModuleDictMixin):
     def get_qualified_names(self):
         return ()
 
-    @property
+    @property  # type: ignore[misc]
     @inference_state_method_cache()
     def name(self):
         string_name = self.py__package__()[-1]
diff --git a/jedi/plugins/pytest.py b/jedi/plugins/pytest.py
index 432385e3..cec23733 100644
--- a/jedi/plugins/pytest.py
+++ b/jedi/plugins/pytest.py
@@ -1,6 +1,6 @@
 from pathlib import Path
 
-from parso.python.tree import search_ancestor
+from parso.tree import search_ancestor
 from jedi.inference.cache import inference_state_method_cache
 from jedi.inference.imports import load_module_from_path
 from jedi.inference.filters import ParserTreeFilter
diff --git a/jedi/utils.py b/jedi/utils.py
index dd529be7..b7e1c1a8 100644
--- a/jedi/utils.py
+++ b/jedi/utils.py
@@ -2,7 +2,7 @@
 Utilities for end-users.
 """
 
-import __main__
+import __main__  # type: ignore[import]
 from collections import namedtuple
 import logging
 import traceback
diff --git a/setup.cfg b/setup.cfg
index 6929ce30..904b5ffe 100644
--- a/setup.cfg
+++ b/setup.cfg
@@ -15,7 +15,40 @@ ignore =
   W503,
   # Single letter loop variables are often fine
   E741,
+per-file-ignores =
+  # Ignore apparently unused imports in files where we're (implicitly)
+  # re-exporting them.
+  jedi/__init__.py:F401
+  jedi/inference/compiled/__init__.py:F401
+  jedi/inference/value/__init__.py:F401
 exclude = jedi/third_party/* .tox/*
 
 [pycodestyle]
 max-line-length = 100
+
+
+[mypy]
+# Ensure generics are explicit about what they are (e.g: `List[str]` rather than
+# just `List`)
+disallow_any_generics = True
+
+disallow_subclassing_any = True
+
+# Avoid creating future gotchas emerging from bad typing
+warn_redundant_casts = True
+warn_unused_ignores = True
+warn_return_any = True
+warn_unused_configs = True
+
+warn_unreachable = True
+
+# Require values to be explicitly re-exported; this makes things easier for
+# Flake8 too and avoids accidentally importing thing from the "wrong" place
+# (which helps avoid circular imports)
+implicit_reexport = False
+
+strict_equality = True
+
+[mypy-jedi,jedi.inference.compiled,jedi.inference.value]
+# Various __init__.py files which contain re-exports we want to implicitly make.
+implicit_reexport = True
diff --git a/setup.py b/setup.py
index 041ae0b2..1f660a4d 100755
--- a/setup.py
+++ b/setup.py
@@ -44,6 +44,7 @@ setup(name='jedi',
           ],
           'qa': [
               'flake8==3.8.3',
+              'mypy==0.782',
           ],
       },
       package_data={'jedi': ['*.pyi', 'third_party/typeshed/LICENSE',
diff --git a/sith.py b/sith.py
index 3b286894..7736ab8a 100755
--- a/sith.py
+++ b/sith.py
@@ -44,7 +44,7 @@ Options:
   --pudb                Launch pudb when error is raised.
 """
 
-from docopt import docopt
+from docopt import docopt  # type: ignore[import]
 
 import json
 import os