diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 1a56259db..7ea14b372 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -23,10 +23,9 @@ jobs:
           - macOS
         python-version:
           - "3.10"
-          - 3.9
-          - 3.6
-          - 3.7
-          - 3.8
+          - "3.9"
+          - "3.8"
+          - "3.7"
         pip-version:
           - "latest"
           - "previous"
diff --git a/.github/workflows/cron.yml b/.github/workflows/cron.yml
index eecb557db..178c3c67e 100644
--- a/.github/workflows/cron.yml
+++ b/.github/workflows/cron.yml
@@ -17,10 +17,10 @@ jobs:
           - Windows
           - MacOS
         python-version:
-          - 3.9
-          - 3.6
-          - 3.7
-          - 3.8
+          - "3.10"
+          - "3.9"
+          - "3.8"
+          - "3.7"
         pip-version:
           - main
     env:
diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index 2be80487b..baeb5767a 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -12,7 +12,7 @@ repos:
     rev: v2.29.0
     hooks:
       - id: pyupgrade
-        args: [--py36-plus]
+        args: [--py37-plus]
   - repo: https://github.com/PyCQA/flake8
     rev: 4.0.1
     hooks:
diff --git a/README.rst b/README.rst
index 3b51277b4..8c4dd7b80 100644
--- a/README.rst
+++ b/README.rst
@@ -454,7 +454,7 @@ then yes, you should commit both ``requirements.in`` and ``requirements.txt`` to
 Note that if you are deploying on multiple Python environments (read the section below),
 then you must commit a seperate output file for each Python environment.
 We suggest to use the ``{env}-requirements.txt`` format
-(ex: ``win32-py3.7-requirements.txt``, ``macos-py3.6-requirements.txt``, etc.).
+(ex: ``win32-py3.7-requirements.txt``, ``macos-py3.10-requirements.txt``, etc.).
 
 
 Cross-environment usage of ``requirements.in``/``requirements.txt`` and ``pip-compile``
@@ -462,7 +462,7 @@ Cross-environment usage of ``requirements.in``/``requirements.txt`` and ``pip-co
 
 The dependencies of a package can change depending on the Python environment in which it
 is installed.  Here, we define a Python environment as the combination of Operating
-System, Python version (3.6, 3.7, etc.), and Python implementation (CPython, PyPy,
+System, Python version (3.7, 3.8, etc.), and Python implementation (CPython, PyPy,
 etc.). For an exact definition, refer to the possible combinations of `PEP 508
 environment markers`_.
 
@@ -527,5 +527,7 @@ versions as the required ``pip`` versions.
 +---------------+----------------+----------------+
 | 6.0.0 - 6.3.1 | 20.3 - 21.2.*  | 3.6 - 3.9      |
 +---------------+----------------+----------------+
-| 6.4.0+        | 21.2+          | 3.6 - 3.10     |
+| 6.4.0         | 21.2 - 21.3.*  | 3.6 - 3.10     |
++---------------+----------------+----------------+
+| 6.5.0+        | 21.2+          | 3.7 - 3.10     |
 +---------------+----------------+----------------+
diff --git a/piptools/_compat/contextlib.py b/piptools/_compat/contextlib.py
deleted file mode 100644
index 8f0c5c863..000000000
--- a/piptools/_compat/contextlib.py
+++ /dev/null
@@ -1,31 +0,0 @@
-# Ported from python 3.7 contextlib.py
-from types import TracebackType
-from typing import Optional, Type, TypeVar
-
-_T = TypeVar("_T")
-
-
-class nullcontext:
-    """Context manager that does no additional processing.
-    Used as a stand-in for a normal context manager, when a particular
-    block of code is only sometimes used with a normal context manager:
-    cm = optional_cm if condition else nullcontext()
-    with cm:
-        # Perform operation, using optional_cm if condition is True
-
-    TODO: replace with `contextlib.nullcontext()` after Python 3.6 being dropped
-    """
-
-    def __init__(self, enter_result: _T) -> None:
-        self.enter_result = enter_result
-
-    def __enter__(self) -> _T:
-        return self.enter_result
-
-    def __exit__(
-        self,
-        exc_type: Optional[Type[BaseException]],
-        exc_val: Optional[BaseException],
-        exc_tb: Optional[TracebackType],
-    ) -> Optional[bool]:
-        pass
diff --git a/piptools/repositories/pypi.py b/piptools/repositories/pypi.py
index 1c59201df..ffaad4187 100644
--- a/piptools/repositories/pypi.py
+++ b/piptools/repositories/pypi.py
@@ -1,3 +1,4 @@
+import contextlib
 import hashlib
 import itertools
 import logging
@@ -39,7 +40,7 @@
 from pip._vendor.packaging.version import _BaseVersion
 from pip._vendor.requests import RequestException, Session
 
-from .._compat import contextlib
+from .._compat import PIP_VERSION
 from ..exceptions import NoCandidateFound
 from ..logging import log
 from ..utils import (
@@ -103,7 +104,8 @@ def __init__(self, pip_args: List[str], cache_dir: str):
         self._cache_dir = normalize_path(str(cache_dir))
         self._download_dir = os.path.join(self._cache_dir, "pkgs")
 
-        self._setup_logging()
+        if PIP_VERSION[0] < 22:
+            self._setup_logging()
 
     def clear_caches(self) -> None:
         rmtree(self._download_dir, ignore_errors=True)
@@ -439,7 +441,8 @@ def _wheel_support_index_min(self: Wheel, tags: List[Tag]) -> int:
     def _setup_logging(self) -> None:
         """
         Setup pip's logger. Ensure pip is verbose same as pip-tools and sync
-        pip's log stream with LogContext.stream.
+        pip's log stream with LogContext.stream. This is only necessary for
+        pip<22.0.
         """
         # Default pip's logger is noisy, so decrease it's verbosity
         setup_logging(
diff --git a/piptools/utils.py b/piptools/utils.py
index 3134a5091..dc254a80e 100644
--- a/piptools/utils.py
+++ b/piptools/utils.py
@@ -1,4 +1,5 @@
 import collections
+import copy
 import itertools
 import json
 import os
@@ -26,6 +27,7 @@
 from pip._internal.vcs import is_url
 from pip._vendor.packaging.markers import Marker
 from pip._vendor.packaging.specifiers import SpecifierSet
+from pip._vendor.packaging.utils import canonicalize_name
 from pip._vendor.packaging.version import Version
 from pip._vendor.pkg_resources import Distribution, Requirement, get_distribution
 
@@ -121,7 +123,11 @@ def format_requirement(
     elif is_url_requirement(ireq):
         line = _build_direct_reference_best_efforts(ireq)
     else:
-        line = str(ireq.req).lower()
+        # Canonicalize the requirement name
+        # https://packaging.pypa.io/en/latest/utils.html#packaging.utils.canonicalize_name
+        req = copy.copy(ireq.req)
+        req.name = canonicalize_name(req.name)
+        line = str(req)
 
     if marker:
         line = f"{line} ; {marker}"
diff --git a/setup.cfg b/setup.cfg
index 54d0ed9ef..9e9215efe 100644
--- a/setup.cfg
+++ b/setup.cfg
@@ -15,7 +15,6 @@ classifiers =
     Programming Language :: Python
     Programming Language :: Python :: 3
     Programming Language :: Python :: 3 :: Only
-    Programming Language :: Python :: 3.6
     Programming Language :: Python :: 3.7
     Programming Language :: Python :: 3.8
     Programming Language :: Python :: 3.9
@@ -25,7 +24,7 @@ classifiers =
     Topic :: System :: Systems Administration
 
 [options]
-python_requires = >=3.6
+python_requires = >=3.7
 setup_requires = setuptools_scm
 packages = find:
 zip_safe = false
diff --git a/tests/test_cli_compile.py b/tests/test_cli_compile.py
index 585165908..63544bb13 100644
--- a/tests/test_cli_compile.py
+++ b/tests/test_cli_compile.py
@@ -610,7 +610,7 @@ def test_url_package(runner, line, dependency, generate_hashes):
         ),
         pytest.param(
             path_to_url(os.path.join(PACKAGES_PATH, "small_fake_with_subdir"))
-            + "#subdirectory=subdir&egg=small_fake_a",
+            + "#subdirectory=subdir&egg=small-fake-a",
             "small-fake-a @ "
             + path_to_url(os.path.join(PACKAGES_PATH, "small_fake_with_subdir"))
             + "#subdirectory=subdir",
diff --git a/tests/test_writer.py b/tests/test_writer.py
index 196cd12ed..7884a4532 100644
--- a/tests/test_writer.py
+++ b/tests/test_writer.py
@@ -381,7 +381,8 @@ def test_write_find_links(writer, find_links, expected_lines):
 
 def test_write_order(writer, from_line):
     """
-    Order of packages should match that of `pip freeze`.
+    Order of packages should match that of `pip freeze`, with the exception
+    that requirement names should be canonicalized.
     """
     writer.emit_header = False
 
@@ -393,7 +394,7 @@ def test_write_order(writer, from_line):
     ]
     expected_lines = [
         "package==5.6",
-        "package_a==0.1",
+        "package-a==0.1",
         "package-b==2.3.4",
         "package2==7.8.9",
     ]
diff --git a/tox.ini b/tox.ini
index fac40a2e3..9206cc5e6 100644
--- a/tox.ini
+++ b/tox.ini
@@ -1,7 +1,7 @@
 [tox]
 envlist =
     # NOTE: keep this in sync with the env list in .github/workflows/ci.yml.
-    py{36,37,38,39,310,311,py3}-pip{previous,latest,main}-coverage
+    py{37,38,39,310,311,py3}-pip{previous,latest,main}-coverage
     checkqa
     readme
 skip_missing_interpreters = True
@@ -11,7 +11,7 @@ extras =
     testing
     coverage: coverage
 deps =
-    pipprevious: pip==21.2.*
+    pipprevious: pip==21.3.*
     piplatest: pip
     pipmain: -e git+https://github.com/pypa/pip.git@main#egg=pip
 setenv =