How do I run 32-bit programs on a 64-bit Debian/Ubuntu?
For current releases
Current Debian and Ubuntu have multiarch support: You can mix x86_32 (i386) and x86_64 (amd64) packages on the same system in a straightforward way. This is known as multiarch support - see Ubuntu or Debian wiki more information.
See warl0ck's answer for a simple, up-to-date answer.
For old releases
In older releases, Debian and Ubuntu ship with a number of 32-bit libraries on amd64. Install the ia32-libs 
package to have a basic set of 32-bit libraries, and possibly other packages that depend on this one. Your 32-bit executables should simply run if you have all the required libraries. For development, install gcc-multilib , and again possibly other packages that depend on it such as g++-multilib. You may find binutils-multiarch useful as well, and ia32-libs-dev on Debian. Pass the -m32 option to gcc to compile for ix86.
Note that uname -m will still show x64_64 if you're running a 64-bit kernel, regardless of what 32-bit user mode components you have installed. Schroot described below takes care of this.
Schroot
This section is a guide to installing a Debian-like distribution “inside” another Linux distribution. It is worded in terms of installing a 32-bit Ubuntu inside a 64-bit Ubuntu, but should apply with minor modifications to other situations, such as installing Debian unstable inside Debian stable or vice versa.
Introduction
The idea is to install an alternate distribution in a subtree and run from that. You can install a 32-bit system on a 64-bit system that way, or a different release of your distribution, or a testing environment with different sets of packages installed.
The chroot command and system call starts a process with a view of the filesystem that's restricted to a subtree of the directory tree. Debian and Ubuntu ship schroot, a utility that wraps around this feature to create a more usable sub-environment.
Install the schroot package (Debian) and the debootstrap package (Debian). Debootstrap is only needed for the installation of the alternate distribution and can be removed afterwards.
Set up schroot
This example describes how to set up a 32-bit Ubuntu 10.04LTS (lucid lynx) alternate environment. A similar setup should work with other releases of Debian and Ubuntu. Create a file /etc/schroot/chroot.d/lucid32 with the following contents:
[lucid32]
description=Ubuntu 10.04LTS 32-bit
directory=/32
type=directory
personality=linux32
users=yourusername
groups=users,admin
The line directory=/32 tells schroot where we'll put the files of the 32-bit installation. The line username=yourusername says the user yourusername will be allowed to use the schroot. The line groups=users,admin says that users in either group will be allowed to use the schroot; you can also put a users=… directive.
Install the new distribution
Create the directory and start populating it with debootstrap. Debootstrap downloads and installs a core set of packages for the specified distribution and architecture.
mkdir /32
debootstrap --arch i386 lucid /32 http://archive.ubuntu.com/ubuntu
You almost have a working system already; what follows is minor enhancements. Schroot automatically overwrites several files in /32/etc when you run it, in particular the DNS configuration in /etc/resolv.conf and the user database in /etc/passwd and other files (this can be overridden, see the documentation). There are a few more files you may want to copy manually once and for all:
cp -p /etc/apt/apt.conf /32/etc/apt/ # for proxy settings
cp -p /etc/apt/sources.list /32/etc/apt/ # for universe, security, etc
cp -p /etc/environment /32/etc/ # for proxy and locale settings
cp -p /etc/sudoers /32/etc/ # for custom sudo settings
There won't be a file /etc/mtab or /etc/fstab in the chroot. I don't recommend using the mount command manually in the chroot, do it from outside. But do create a good-enough /etc/mtab to make commands such as df work reasonably.
ln -s /proc/mounts /32/etc/mtab
With the directory type, schroot will perform bind mounts of a number of directories, i.e. those directories will be shared with the parent installation: /proc, /dev, /home, /tmp.
Services in the chroot
As described here, a schroot is not suitable for running daemons. Programs in the schroot will be killed when you exit the schroot. Use a “plain” schroot instead of a “directory” schroot if you want it to be more permanent, and set up permanent bind mounts in /etc/fstab on the parent installation.
On Debian and Ubuntu, services start automatically on installation. To avoid this (which could disrupt the services running outside the chroot, in particular because network ports are shared), establish a policy of not running services in the chroot. Put the following script as /32/usr/sbin/policy-rc.d and make it executable (chmod a+rx /32/usr/sbin/policy-rc.d).
#!/bin/sh
## Don't start any service if running in a chroot.
## See /usr/share/doc/sysv-rc/README.policy-rc.d.gz
if [ "$(stat -c %d:%i /)" != "$(stat -c %d:%i /proc/1/root/.)" ]; then
exit 101
fi
Populate the new system
Now we can start using the chroot. You'll want to install a few more packages at this point.
schroot -c lucid32
sudo apt-get update
apt-get install lsb-core nano
...
You may need to generate a few locales, e.g.
locale-gen en_US en_US.utf8
If the schroot is for an older release of Ubuntu such as 8.04 (hardy), note that the package ubuntu-standard pulls in an MTA. Select nullmailer instead of the default postfix (you may want your chroot to send mail but you definitely don't want it to receive any).
Going further
For more information, see the schroot manual, the schroot FAQ and the
schroot.conf manual. Schroot is part of the Debian autobuilder (buildd) project. There may be additional useful tips on the Ubuntu community page about debootstrap.
Virtual machine
If you need complete isolation of the alternate environment, use a virtual machine such as KVM (qemu-kvm ) or VirtualBox.
Since Ubuntu 11.04 (natty) and Debian 7.0 (wheezy) introduced multiarch support, 32-bit and 64-bit libraries can coexist on one system. To install a 32-bit library libXX, first add the necessary 32bit architecture to your system:
sudo dpkg --add-architecture i386
Then install the 32bit library:
sudo apt-get install libXX:i386
The ia32-libs package. As of Ubuntu 12.04 precise, it no longer contains any libraries, it only pulls in libXX:i386 packages as dependencies.
The compile part is rather easy for C and C++ programs, add -m32 to CFLAG or CXXFLAG, which will make generated program 32bit, e.g
export CFLAGS="-m32"
gcc main.c -o main
This works for makefile-based projects as well.
If you just have a 32-bit binary you'd like to run on a modern 64-bit Debian/Ubuntu system, do the following:
dpkg --add-architecture i386
apt update
apt install libc6-i386
This has been tested on Debian 9.