diff --git a/.travis.yml b/.travis.yml
index bcbf75a43..d6ec88e06 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -17,12 +17,16 @@ matrix:
       python: 3.7  # Keep in sync with .readthedocs.yml
 
     - env: TOXENV=pypy3
+    - env: TOXENV=pinned
+      python: 3.5.1
+      dist: trusty
+    - env: TOXENV=asyncio
+      python: 3.5.1  # We use additional code to support 3.5.3 and earlier
+      dist: trusty
     - env: TOXENV=py
       python: 3.5
-    - env: TOXENV=pinned
-      python: 3.5
     - env: TOXENV=asyncio
-      python: 3.5.2
+      python: 3.5  # We use specific code to support >= 3.5.4, < 3.6
     - env: TOXENV=py
       python: 3.6
     - env: TOXENV=py
diff --git a/README.rst b/README.rst
index ce5973bcd..fd84e127e 100644
--- a/README.rst
+++ b/README.rst
@@ -40,7 +40,7 @@ including a list of features.
 Requirements
 ============
 
-* Python 3.5+
+* Python 3.5.1+
 * Works on Linux, Windows, macOS, BSD
 
 Install
diff --git a/docs/faq.rst b/docs/faq.rst
index 75a0f4864..936f315b3 100644
--- a/docs/faq.rst
+++ b/docs/faq.rst
@@ -69,7 +69,7 @@ Here's an example spider using BeautifulSoup API, with ``lxml`` as the HTML pars
 What Python versions does Scrapy support?
 -----------------------------------------
 
-Scrapy is supported under Python 3.5+
+Scrapy is supported under Python 3.5.1+
 under CPython (default Python implementation) and PyPy (starting with PyPy 5.9).
 Python 3 support was added in Scrapy 1.1.
 PyPy support was added in Scrapy 1.4, PyPy3 support was added in Scrapy 1.5.
diff --git a/docs/intro/install.rst b/docs/intro/install.rst
index 6356e0eea..4af80d801 100644
--- a/docs/intro/install.rst
+++ b/docs/intro/install.rst
@@ -7,7 +7,7 @@ Installation guide
 Installing Scrapy
 =================
 
-Scrapy runs on Python 3.5 or above under CPython (default Python
+Scrapy runs on Python 3.5.1 or above under CPython (default Python
 implementation) and PyPy (starting with PyPy 5.9).
 
 If you're using `Anaconda`_ or `Miniconda`_, you can install the package from
diff --git a/tests/test_proxy_connect.py b/tests/test_proxy_connect.py
index 4763a5417..eb4ecc91d 100644
--- a/tests/test_proxy_connect.py
+++ b/tests/test_proxy_connect.py
@@ -4,6 +4,7 @@ import re
 import sys
 from subprocess import Popen, PIPE
 from urllib.parse import urlsplit, urlunsplit
+from unittest import skipIf
 
 import pytest
 from testfixtures import LogCapture
@@ -56,6 +57,8 @@ def _wrong_credentials(proxy_url):
     return urlunsplit(bad_auth_proxy)
 
 
+@skipIf(sys.version_info < (3, 5, 4),
+        "requires mitmproxy < 3.0.0, which these tests do not support")
 class ProxyConnectTestCase(TestCase):
 
     def setUp(self):
@@ -80,7 +83,7 @@ class ProxyConnectTestCase(TestCase):
             yield crawler.crawl(self.mockserver.url("/status?n=200", is_secure=True))
         self._assert_got_response_code(200, log)
 
-    @pytest.mark.xfail(reason='Python 3.6+ fails this earlier', condition=sys.version_info.minor >= 6)
+    @pytest.mark.xfail(reason='Python 3.6+ fails this earlier', condition=sys.version_info >= (3, 6))
     @defer.inlineCallbacks
     def test_https_connect_tunnel_error(self):
         crawler = get_crawler(SimpleSpider)