diff --git a/contrib/packaging/debian/rules b/contrib/packaging/debian/rules
--- a/contrib/packaging/debian/rules
+++ b/contrib/packaging/debian/rules
@@ -2,14 +2,51 @@
 # Uncomment this to turn on verbose mode.
 # export DH_VERBOSE=1
 
+# By default we build a .deb where the native components are built with the
+# current "default" version of py3 on the build machine. If you wish to build a
+# .deb that has native components built for multiple versions of py3:
+#
+#   1. install python3.x and python3.x-dev for each version you want
+#   2. set DEB_HG_MULTI_VERSION=1 or DEB_HG_PYTHON_VERSIONS in your environment
+#      (if both are set, DEB_HG_PYTHON_VERSIONS has precedence)
+#
+# If you choose `DEB_HG_MULTI_VERSION=1`, it will build for every "supported"
+# version of py3 that's installed on the build machine. This may not be equal to
+# the actual versions that are installed, see the comment above where we set
+# DEB_HG_PYTHON_VERSIONS below. If you choose to set `DEB_HG_PYTHON_VERSIONS`
+# yourself, set it to a space-separated string of python version numbers, like:
+#   DEB_HG_PYTHON_VERSIONS="3.7 3.8" make deb
+DEB_HG_MULTI_VERSION?=0
+
 CPUS=$(shell cat /proc/cpuinfo | grep -E ^processor | wc -l)
 
+# By default, only build for the version of python3 that the system considers
+# the 'default' (which should be the one invoked by just running 'python3'
+# without a minor version). If DEB_HG_PYTHON_VERSIONS is set, this is ignored.
+ifeq ($(DEB_HG_MULTI_VERSION), 1)
+	# If we're building for multiple versions, use all of the "supported" versions
+	# on the build machine. Note: the mechanism in use here (`py3versions`) is the
+	# recommended one, but it relies on a file written by the python3-minimal
+	# package, and this file is not dynamic and does not account for manual
+	# installations, just the ones that would be installed by `python3-all`. This
+	# includes the `-i` flag, which claims it's to list all "installed" versions,
+	# but it doesn't. This was quite confusing, hence this tale of woe. :)
+	DEB_HG_PYTHON_VERSIONS?=$(shell py3versions -vs)
+else
+	# If we're building for only one version, identify the "default" version on
+	# the build machine and use that when building; this is just so that we don't
+	# have to duplicate the rules below for multi-version vs. single-version. The
+	# shebang line will still be /usr/bin/python3 (no minor version).
+	DEB_HG_PYTHON_VERSIONS?=$(shell py3versions -vd)
+endif
+
 export HGPYTHON3=1
 export PYTHON=python3
 
 %:
 	dh $@ --with python3
 
+# Note: testing can be disabled using the standard `DEB_BUILD_OPTIONS=nocheck`
 override_dh_auto_test:
 	http_proxy='' dh_auto_test -- TESTFLAGS="-j$(CPUS)"
 
@@ -24,8 +61,15 @@ override_dh_auto_build:
 	$(MAKE) all
 	$(MAKE) -C contrib/chg all
 
-override_dh_auto_install:
-	python3 setup.py install --root "$(CURDIR)"/debian/mercurial --install-layout=deb
+# Build the native extensions for a specfic python3 version (which must be
+# installed on the build machine).
+install-python%:
+	python$* setup.py install --root "$(CURDIR)"/debian/mercurial --install-layout=deb
+
+# Build the final package. This rule has a dependencies section that causes the
+# native extensions to be compiled for every version of python3 listed in
+# DEB_HG_PYTHON_VERSIONS.
+override_dh_auto_install: $(DEB_HG_PYTHON_VERSIONS:%=install-python%)
 	# chg
 	make -C contrib/chg \
 		DESTDIR="$(CURDIR)"/debian/mercurial \