diff --git a/.appveyor.yml b/.appveyor.yml
index f0a0dd3..816f33e 100644
--- a/.appveyor.yml
+++ b/.appveyor.yml
@@ -16,14 +16,6 @@ environment:
     - PYTHON: "C:\\Python27-x64"
       DISTUTILS_USE_SDK: "1"
       TOXENV: py27-nodilate
-    - PYTHON: "C:\\Python35"
-      TOXENV: py35
-    - PYTHON: "C:\\Python35-x64"
-      TOXENV: py35
-    - PYTHON: "C:\\Python36"
-      TOXENV: py36
-    - PYTHON: "C:\\Python36-x64"
-      TOXENV: py36
     - PYTHON: "C:\\Python37-x64"
       TOXENV: py37
 
diff --git a/.github/workflows/test.yml b/.github/workflows/test.yml
index 5ef8405..fef66e2 100644
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -11,7 +11,7 @@ jobs:
     runs-on: ubuntu-latest
     strategy:
       matrix:
-        python-version: [3.5, 3.6, 3.7, 3.8, 3.9]
+        python-version: [3.7, 3.8, 3.9]
 
     steps:
     - uses: actions/checkout@v2
diff --git a/NEWS.md b/NEWS.md
index a142bfa..1cf5f78 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,5 +1,9 @@
 User-visible changes in "magic-wormhole":
 
+## Upcoming Release
+
+* Python 3.5 and 3.6 are past their EOL date and support is dropped (#448)
+
 ## Release 0.12.0 (04-Apr-2020)
 
 * A command like `wormhole send /dev/fd0` can send the contents of the named
diff --git a/README.md b/README.md
index 72d3099..3b2cb52 100644
--- a/README.md
+++ b/README.md
@@ -37,10 +37,10 @@ and the
 
 Magic-Wormhole is released under the MIT license, see the `LICENSE` file for details.
 
-This library is compatible with Python 3.6 and higher (tested against 3.6,
-3.7, 3.8, and 3.9). It also still works with Python 2.7 and 3.5, although
-these are no longer supported by upstream libraries like Cryptography, so it
-may stop working at any time.
+This library is compatible with Python 3.8 and higher (tested against
+3.8, and 3.9). It also still works with Python 2.7 although that is no
+longer supported by upstream libraries like Cryptography, so it may
+stop working at any time.
 
 ## Packaging, Installation
 
diff --git a/docs/welcome.md b/docs/welcome.md
index 6285d92..851774a 100644
--- a/docs/welcome.md
+++ b/docs/welcome.md
@@ -312,8 +312,7 @@ trial wormhole.test.test_cli.PregeneratedCode.test_file_tor
 ```
 
 Developers can also just clone the source tree and run `tox` to run the unit
-tests on all supported (and installed) versions of python: 2.7, 3.4, 3.5, and
-3.6.
+tests on all supported (and installed) versions of python: 2.7, 3.7 and 3.8.
 
 ### Troubleshooting
 
@@ -336,8 +335,7 @@ Relevant [xkcd](https://xkcd.com/949/) :-)
 
 This library is released under the MIT license, see LICENSE for details.
 
-This library is compatible with python2.7, 3.4 (non-Windows-only), 3.5, and
-3.6 .
+This library is compatible with python2.7, 3.7 and 3.8 .
 
 
 <!-- footnotes -->
diff --git a/setup.py b/setup.py
index 2178388..248cdc7 100644
--- a/setup.py
+++ b/setup.py
@@ -11,8 +11,6 @@ trove_classifiers = [
     "Programming Language :: Python :: 2",
     "Programming Language :: Python :: 2.7",
     "Programming Language :: Python :: 3",
-    "Programming Language :: Python :: 3.5",
-    "Programming Language :: Python :: 3.6",
     "Programming Language :: Python :: 3.7",
     "Programming Language :: Python :: 3.8",
     "Programming Language :: Python :: Implementation :: CPython",
diff --git a/tox.ini b/tox.ini
index 914b22e..e1e6e8b 100644
--- a/tox.ini
+++ b/tox.ini
@@ -4,8 +4,8 @@
 # and then run "tox" from this directory.
 
 [tox]
-# useful envs: py27-nodilate, py35, py36, py37, py38, pypy, flake8
-envlist = {py27-nodilate,py35,py36,py37,py38}
+# useful envs: py27-nodilate, py37, py38, pypy, flake8
+envlist = {py27-nodilate,py37,py38}
 skip_missing_interpreters = True
 minversion = 2.4.0