diff --git a/.github/workflows/tests.yaml b/.github/workflows/tests.yaml
index c27ae784ea8..9d6b6a526c4 100644
--- a/.github/workflows/tests.yaml
+++ b/.github/workflows/tests.yaml
@@ -10,11 +10,7 @@ jobs:
       fail-fast: false
       matrix:
         os: [ubuntu-latest, windows-latest, macos-latest]
-        python-version: ["3.6", "3.7", "3.8", "3.9"]
-        exclude:
-          # Several failures; Python 3.6 to be phased out soon
-          - os: macos-latest
-            python-version: "3.6"
+        python-version: ["3.7", "3.8", "3.9"]
 
         # Uncomment to stress-test the test suite for random failures
         # This will take a LONG time and delay all PRs across the whole github.com/dask!
@@ -23,6 +19,8 @@ jobs:
     steps:
       - name: Checkout source
         uses: actions/checkout@v2
+        with:
+          fetch-depth: 0
 
       - name: Setup Conda Environment
         uses: conda-incubator/setup-miniconda@v2
diff --git a/continuous_integration/environment-3.6.yaml b/continuous_integration/environment-3.6.yaml
deleted file mode 100644
index 7a13f63b232..00000000000
--- a/continuous_integration/environment-3.6.yaml
+++ /dev/null
@@ -1,47 +0,0 @@
-name: dask-distributed
-channels:
-  - conda-forge
-  - defaults
-dependencies:
-  - python=3.6
-  - pip
-  - asyncssh
-  - bokeh
-  - click
-  - cloudpickle
-  - contextvars  # Backport from Python 3.7
-  - dask  # overridden by git tip below
-  - filesystem-spec
-  - h5py
-  - ipykernel
-  - ipywidgets
-  - joblib
-  - jupyter_client
-  - msgpack-python
-  - netcdf4
-  - paramiko
-  - pickle5  # Backport from Python 3.8
-  - prometheus_client
-  - psutil
-  - pytest
-  - pytest-asyncio<0.14.0
-  - pytest-faulthandler
-  - pytest-repeat
-  - pytest-rerunfailures
-  - pytest-timeout
-  - requests
-  - s3fs
-  - scikit-learn
-  - scipy
-  - sortedcollections
-  - tblib
-  - toolz
-  - tornado=5  # Only tested here
-  - zict
-  - zstandard
-  - pip:
-      # The '.git@master' is a hack needed to work around the failure
-      #     ERROR: Package 'dask' requires a different Python: 3.6.12 not in '>=3.6'
-      # It only happens in CI; cannot reproduce locally
-      - git+https://github.com/dask/dask.git@master
-      - keras
diff --git a/continuous_integration/environment-3.7.yaml b/continuous_integration/environment-3.7.yaml
index 7057155b470..bb7838fe205 100644
--- a/continuous_integration/environment-3.7.yaml
+++ b/continuous_integration/environment-3.7.yaml
@@ -36,7 +36,7 @@ dependencies:
   - sortedcollections
   - tblib
   - toolz
-  - tornado=6
+  - tornado=5 # Only tested here
   - zict
   - zstandard
   - pip:
diff --git a/distributed/client.py b/distributed/client.py
index fa17a6b710d..4f1f117f3c5 100644
--- a/distributed/client.py
+++ b/distributed/client.py
@@ -756,15 +756,6 @@ def as_current(self):
         method to return self. Any Future objects deserialized inside this context
         manager will be automatically attached to this Client.
         """
-        # In Python 3.6, contextvars are thread-local but not Task-local.
-        # We can still detect a race condition though.
-        if sys.version_info < (3, 7) and _current_client.get() not in (self, None):
-            raise RuntimeError(
-                "Detected race condition where multiple asynchronous clients tried "
-                "entering the as_current() context manager at the same time. "
-                "Please upgrade to Python 3.7+."
-            )
-
         tok = _current_client.set(self)
         try:
             yield
diff --git a/distributed/comm/tests/test_comms.py b/distributed/comm/tests/test_comms.py
index b7a2061af48..750c30d5c27 100644
--- a/distributed/comm/tests/test_comms.py
+++ b/distributed/comm/tests/test_comms.py
@@ -223,8 +223,7 @@ async def handle_comm(comm):
     assert not handle_comm_called
 
     writer.close()
-    if hasattr(writer, "wait_closed"):  # always true for python >= 3.7, but not for 3.6
-        await writer.wait_closed()
+    await writer.wait_closed()
 
 
 @pytest.mark.asyncio
diff --git a/distributed/compatibility.py b/distributed/compatibility.py
index 6141f3559a7..982b7017951 100644
--- a/distributed/compatibility.py
+++ b/distributed/compatibility.py
@@ -11,16 +11,3 @@
 MACOS = sys.platform == "darwin"
 WINDOWS = sys.platform.startswith("win")
 TORNADO6 = tornado.version_info[0] >= 6
-PY37 = sys.version_info[:2] >= (3, 7)
-
-if sys.version_info[:2] >= (3, 7):
-    from asyncio import get_running_loop
-else:
-
-    def get_running_loop():
-        from asyncio import _get_running_loop
-
-        loop = _get_running_loop()
-        if loop is None:
-            raise RuntimeError("no running event loop")
-        return loop
diff --git a/distributed/deploy/tests/test_local.py b/distributed/deploy/tests/test_local.py
index 5a35f36695c..8eaea2ba83f 100644
--- a/distributed/deploy/tests/test_local.py
+++ b/distributed/deploy/tests/test_local.py
@@ -1020,9 +1020,6 @@ async def test_capture_security(cleanup, temporary):
 
 
 @pytest.mark.asyncio
-@pytest.mark.skipif(
-    sys.version_info < (3, 7), reason="asyncio.all_tasks not implemented"
-)
 async def test_no_danglng_asyncio_tasks(cleanup):
     start = asyncio.all_tasks()
     async with LocalCluster(asynchronous=True, processes=False):
diff --git a/distributed/metrics.py b/distributed/metrics.py
index 0f7d78a8129..163a982b792 100755
--- a/distributed/metrics.py
+++ b/distributed/metrics.py
@@ -84,60 +84,13 @@ def resync(self):
     # Under modern Unices, time.time() should be good enough
     time = timemod.time
 
-
-def _native_thread_time():
-    # Python 3.7+, not all platforms
-    return timemod.thread_time()
-
-
-def _linux_thread_time():
-    # Use hardcoded CLOCK_THREAD_CPUTIME_ID on Python 3 <= 3.6
-    if sys.platform != "linux":
-        raise OSError
-    return timemod.clock_gettime(3)
-
-
-def _native_process_time():
-    # Python 3, should work everywhere
-    return timemod.process_time()
-
-
-def _native_clock_func():
-    # time.clock() unfortunately has different semantics depending on the
-    # platform.  On POSIX it's a per-process CPU timer (with possibly
-    # poor resolution).  On Windows it's a high-resolution wall clock timer.
-    return timemod.clock()
-
-
-def _detect_process_time():
-    """
-    Return a per-process CPU timer function if possible, otherwise
-    a wall-clock timer.
-    """
-    for func in [_native_process_time]:
-        try:
-            func()
-            return func
-        except (AttributeError, OSError):
-            pass
-    # Only Python 2?
-    return _native_clock_func
-
-
-def _detect_thread_time():
-    """
-    Return a per-thread CPU timer function if possible, otherwise
-    a per-process CPU timer function, or at worse a wall-clock timer.
-    """
-    for func in [_native_thread_time, _linux_thread_time, _native_process_time]:
-        try:
-            func()
-            return func
-        except (AttributeError, OSError):
-            pass
-    # Only Python 2?
-    return time
-
-
-process_time = _detect_process_time()
-thread_time = _detect_thread_time()
+process_time = timemod.process_time
+
+# Get a per-thread CPU timer function if possible, otherwise
+# use a per-process CPU timer function.
+try:
+    # thread_time is supported on Python 3.7+ but not all platforms
+    thread_time = timemod.thread_time
+except (AttributeError, OSError):
+    # process_time is supported on Python 3.3+ everywhere
+    thread_time = process_time
diff --git a/distributed/pubsub.py b/distributed/pubsub.py
index 3b580d6180a..91200be06eb 100644
--- a/distributed/pubsub.py
+++ b/distributed/pubsub.py
@@ -1,6 +1,5 @@
 import asyncio
 from collections import defaultdict, deque
-from contextlib import suppress
 import logging
 import threading
 import weakref
@@ -421,8 +420,7 @@ async def _():
             try:
                 await asyncio.wait_for(_(), timeout2)
             finally:
-                with suppress(RuntimeError):  # Python 3.6 fails here sometimes
-                    self.condition.release()
+                self.condition.release()
 
         return self.buffer.popleft()
 
diff --git a/distributed/tests/test_client.py b/distributed/tests/test_client.py
index 100b52eaf81..9caa3ccb08f 100644
--- a/distributed/tests/test_client.py
+++ b/distributed/tests/test_client.py
@@ -3668,7 +3668,6 @@ async def test_reconnect_timeout(c, s):
 @pytest.mark.avoid_ci(reason="hangs on github actions ubuntu-latest CI")
 @pytest.mark.slow
 @pytest.mark.skipif(WINDOWS, reason="num_fds not supported on windows")
-@pytest.mark.skipif(sys.version_info < (3, 7), reason="TODO: intermittent failures")
 @pytest.mark.parametrize("worker,count,repeat", [(Worker, 100, 5), (Nanny, 10, 20)])
 def test_open_close_many_workers(loop, worker, count, repeat):
     psutil = pytest.importorskip("psutil")
@@ -4054,10 +4053,6 @@ def run2():
     t2.join()
 
 
-@pytest.mark.xfail(
-    sys.version_info < (3, 7),
-    reason="Python 3.6 contextvars are not copied on Task creation",
-)
 @gen_cluster(client=False)
 async def test_as_current_is_task_local(s, a, b):
     l1 = asyncio.Lock()
diff --git a/distributed/tests/test_core.py b/distributed/tests/test_core.py
index e9b77c0b1a3..071b6b039c7 100644
--- a/distributed/tests/test_core.py
+++ b/distributed/tests/test_core.py
@@ -1,7 +1,6 @@
 import asyncio
 import os
 import socket
-import sys
 import threading
 import weakref
 import warnings
@@ -180,10 +179,6 @@ class MyServer(Server):
     default_port = 8756
 
 
-@pytest.mark.skipif(
-    sys.version_info < (3, 7),
-    reason="asynccontextmanager not avaiable before Python 3.7",
-)
 @pytest.mark.asyncio
 async def test_server_listen():
     """
diff --git a/distributed/tests/test_diskutils.py b/distributed/tests/test_diskutils.py
index f4afb6d66d4..3b97fde1498 100644
--- a/distributed/tests/test_diskutils.py
+++ b/distributed/tests/test_diskutils.py
@@ -277,8 +277,6 @@ def _test_workspace_concurrency(tmpdir, timeout, max_procs):
 def test_workspace_concurrency(tmpdir):
     if WINDOWS:
         raise pytest.xfail.Exception("TODO: unknown failure on windows")
-    if sys.version_info < (3, 7):
-        raise pytest.xfail.Exception("TODO: unknown failure on Python 3.6")
     _test_workspace_concurrency(tmpdir, 5.0, 6)
 
 
diff --git a/distributed/tests/test_scheduler.py b/distributed/tests/test_scheduler.py
index 0d4cad348c2..701a874fbcc 100644
--- a/distributed/tests/test_scheduler.py
+++ b/distributed/tests/test_scheduler.py
@@ -1818,9 +1818,6 @@ async def test_get_task_duration(c, s, a, b):
 
 
 @pytest.mark.asyncio
-@pytest.mark.skipif(
-    sys.version_info < (3, 7), reason="asyncio.all_tasks not implemented"
-)
 async def test_no_danglng_asyncio_tasks(cleanup):
     start = asyncio.all_tasks()
     async with Scheduler(port=0) as s:
diff --git a/distributed/utils.py b/distributed/utils.py
index dd6ca2fad02..115e9578b6b 100644
--- a/distributed/utils.py
+++ b/distributed/utils.py
@@ -55,7 +55,7 @@
 except ImportError:
     PollIOLoop = None  # dropped in tornado 6.0
 
-from .compatibility import PYPY, WINDOWS, get_running_loop
+from .compatibility import PYPY, WINDOWS
 from .metrics import time
 
 
@@ -1165,7 +1165,7 @@ def reset_logger_locks():
 
     if is_kernel():
         try:
-            get_running_loop()
+            asyncio.get_running_loop()
         except RuntimeError:
             is_kernel_and_no_running_loop = True
 
diff --git a/distributed/variable.py b/distributed/variable.py
index c3fdc94d0d7..83bea566b45 100644
--- a/distributed/variable.py
+++ b/distributed/variable.py
@@ -89,8 +89,7 @@ async def _():  # Python 3.6 is odd and requires special help here
 
                 await asyncio.wait_for(_(), timeout=left)
             finally:
-                with suppress(RuntimeError):  # Python 3.6 loses lock on finally clause
-                    self.started.release()
+                self.started.release()
 
         record = self.variables[name]
         if record["type"] == "Future":
diff --git a/docs/source/install.rst b/docs/source/install.rst
index db1bf316400..695cd3e777f 100644
--- a/docs/source/install.rst
+++ b/docs/source/install.rst
@@ -38,6 +38,6 @@ Notes
 <https://trac.macports.org/ticket/50058>`_.  with Python from macports that
 makes executables be placed in a location that is not available by default. A
 simple solution is to extend the ``PATH`` environment variable to the location
-where Python from macports install the binaries. For example, for Python 3.6::
+where Python from macports install the binaries. For example, for Python 3.7::
 
-    $ export PATH=/opt/local/Library/Frameworks/Python.framework/Versions/3.6/bin:$PATH
+    $ export PATH=/opt/local/Library/Frameworks/Python.framework/Versions/3.7/bin:$PATH
diff --git a/requirements.txt b/requirements.txt
index 37d23c45e69..fd8159cfefe 100644
--- a/requirements.txt
+++ b/requirements.txt
@@ -1,6 +1,5 @@
 click >= 6.6
 cloudpickle >= 1.5.0
-contextvars;python_version<'3.7'
 dask >= 2021.03.0
 msgpack >= 0.6.0
 psutil >= 5.0
diff --git a/setup.py b/setup.py
index 5483e0b963d..8aaaeee5de8 100755
--- a/setup.py
+++ b/setup.py
@@ -67,7 +67,7 @@
     url="https://distributed.dask.org",
     maintainer="Matthew Rocklin",
     maintainer_email="mrocklin@gmail.com",
-    python_requires=">=3.6",
+    python_requires=">=3.7",
     license="BSD",
     package_data={
         "": ["templates/index.html", "template.html"],
@@ -89,7 +89,6 @@
         "License :: OSI Approved :: BSD License",
         "Operating System :: OS Independent",
         "Programming Language :: Python",
-        "Programming Language :: Python :: 3.6",
         "Programming Language :: Python :: 3.7",
         "Programming Language :: Python :: 3.8",
         "Programming Language :: Python :: 3.9",