diff --git a/README.rst b/README.rst
index a54d1bb..bac4598 100644
--- a/README.rst
+++ b/README.rst
@@ -51,9 +51,9 @@ All the following commands are equivalent and run the PyPy 2.7, version
 
 - ``/opt/pypy/pypy2.7-7.3.0/bin/python``
 
-- ``/opt/python/pp273-pypy_73/bin/pypy``
+- ``/opt/python/pp27-pypy_73/bin/pypy``
 
-- ``/opt/python/pp273-pypy_73/bin/python``
+- ``/opt/python/pp27-pypy_73/bin/python``
 
 Similarly, these are the commands to run PyPy 3.6, version 7.3.0:
 
@@ -61,23 +61,28 @@ Similarly, these are the commands to run PyPy 3.6, version 7.3.0:
 
 - ``/opt/pypy/pypy3.6-7.3.0/bin/python``
 
-- ``/opt/python/pp373-pypy36_pp73/bin/pypy``
+- ``/opt/python/pp36-pypy36_pp73/bin/pypy``
 
-- ``/opt/python/pp373-pypy36_pp73/bin/python``
+- ``/opt/python/pp36-pypy36_pp73/bin/python``
 
 
 PEP 425 Compatibility tags
 ---------------------------
 
-``pp273-pypy_73`` and ``pp373-pypy36_pp73`` are the `PEP 425`_ compliant
+``pp27-pypy_73`` and ``pp36-pypy36_pp73`` are the `PEP 425`_ compliant
 compatibility tag. In particular:
 
 - ``pp`` stands for PyPy (as opposed to ``cp`` which is CPython)
 
-- ``273`` and ``373`` mean "Python [2|3]", "PyPy 7.3.x".
+- ``27`` and ``36`` mean "Python 2.7|3.6"
 
 - ``pypy_73`` and ``pypy36_pp73`` (or before PyPy 7.3.0, ``pypy_41`` and
   ``pypy3_71``) are the binary ABI tags for the relevant version of PyPy.
   You can probably ignore them.
 
+Before pip 20 and wheel 0.34, tags looked like ``pp273-pypy_73`` or
+``pp373-pypy36_pp73``, where ``273`` and ``373`` mean  "Python [2|3]",
+"PyPy 7.3.x". Pre-7.3.0 versions of PyPy do not support pip>=20, and thus
+still rely on these old PyPy PEP 425 tags.
+
 .. _`PEP 425`: https://www.python.org/dev/peps/pep-0425/
diff --git a/docker/build_scripts/requirements-pre-7.3.0.txt b/docker/build_scripts/requirements-pre-7.3.0.txt
new file mode 100644
index 0000000..568c084
--- /dev/null
+++ b/docker/build_scripts/requirements-pre-7.3.0.txt
@@ -0,0 +1,9 @@
+pip==19.1.1 \
+    --hash=sha256:44d3d7d3d30a1eb65c7e5ff1173cdf8f7467850605ac7cc3707b6064bddd0958 \
+    --hash=sha256:993134f0475471b91452ca029d4390dc8f298ac63a712814f101cd1b6db46676
+wheel==0.31.1 \
+    --hash=sha256:80044e51ec5bbf6c894ba0bc48d26a8c20a9ba629f4ca19ea26ecfcf87685f5f \
+    --hash=sha256:0a2e54558a0628f2145d2fc822137e322412115173e8a2ddbe1c9024338ae83c
+setuptools==41.0.1 \
+    --hash=sha256:a222d126f5471598053c9a77f4b5d4f26eaa1f150ad6e01dcf1a42e185d05613 \
+    --hash=sha256:c7769ce668c7a333d84e17fe8b524b1c45e7ee9f7908ad0a73e1eda7e6a5aebf
diff --git a/docker/build_scripts/requirements.txt b/docker/build_scripts/requirements.txt
index 2a62061..237047a 100644
--- a/docker/build_scripts/requirements.txt
+++ b/docker/build_scripts/requirements.txt
@@ -1,11 +1,14 @@
 # pip requirements for all cpythons
 # NOTE: pip has GPG signatures; could download and verify independently.
-pip==19.1.1 \
-    --hash=sha256:44d3d7d3d30a1eb65c7e5ff1173cdf8f7467850605ac7cc3707b6064bddd0958 \
-    --hash=sha256:993134f0475471b91452ca029d4390dc8f298ac63a712814f101cd1b6db46676
-wheel==0.31.1 \
-    --hash=sha256:80044e51ec5bbf6c894ba0bc48d26a8c20a9ba629f4ca19ea26ecfcf87685f5f \
-    --hash=sha256:0a2e54558a0628f2145d2fc822137e322412115173e8a2ddbe1c9024338ae83c
-setuptools==41.0.1 \
-    --hash=sha256:a222d126f5471598053c9a77f4b5d4f26eaa1f150ad6e01dcf1a42e185d05613 \
-    --hash=sha256:c7769ce668c7a333d84e17fe8b524b1c45e7ee9f7908ad0a73e1eda7e6a5aebf
+pip==20.0.2 \
+    --hash=sha256:4ae14a42d8adba3205ebeb38aa68cfc0b6c346e1ae2e699a0b3bad4da19cef5c \
+    --hash=sha256:7db0c8ea4c7ea51c8049640e8e6e7fde949de672bfa4949920675563a5a6967f
+wheel==0.34.2 \
+    --hash=sha256:8788e9155fe14f54164c1b9eb0a319d98ef02c160725587ad60f14ddc57b6f96 \
+    --hash=sha256:df277cb51e61359aba502208d680f90c0493adec6f0e848af94948778aed386e
+setuptools==44.0.0; python_version == '2.7' \
+    --hash=sha256:180081a244d0888b0065e18206950d603f6550721bd6f8c0a10221ed467dd78e \
+    --hash=sha256:e5baf7723e5bb8382fc146e33032b241efc63314211a3a120aaa55d62d2bb008
+setuptools==45.2.0; python_version >= '3.5' \
+    --hash=sha256:316484eebff54cc18f322dea09ed031b7e3eb00811b19dcedb09bc09bba7d93d \
+    --hash=sha256:89c6e6011ec2f6d57d43a3f9296c4ef022c2cbf49bab26b407fe67992ae3397f
diff --git a/docker/build_scripts_pypy/install_pypy.sh b/docker/build_scripts_pypy/install_pypy.sh
index c3706d9..40096ac 100755
--- a/docker/build_scripts_pypy/install_pypy.sh
+++ b/docker/build_scripts_pypy/install_pypy.sh
@@ -7,6 +7,11 @@ function get_shortdir {
     $exe -c 'import sys; print("pypy%d.%d-%d.%d.%d" % (sys.version_info[:2]+sys.pypy_version_info[:3]))'
 }
 
+function get_requirements_txt {
+    local exe=$1
+    $exe -c 'import sys; print("requirements-pre-7.3.0.txt" if sys.pypy_version_info < (7,3,0) else "requirements.txt")'
+}
+
 # this is more or less equivalent to do_cpython_build, although we download a
 # prebuilt pypy instead of building it from scratch
 function install_one_pypy {