There are Linux builds of Firefox Aurora available on Mozilla. But if you’re on Debian testing (or like to live dangerously with unstable) and prefer installing/updating via APT, then you’d might want to install Iceweasel Aurora.

I’m on Debian testing and this is what I did:

  1. Run apt-get install debian-keyring and apt-get install pkg-mozilla-archive-keyring (so that APT has signed keys and won’t whine when you update)
  2. Edit the file /etc/apt/sources.list and stick in this line deb http://mozilla.debian.net/ experimental iceweasel-aurora
  3. Run apt-get update, followed by apt-get install -t experimental iceweasel-aurora to install Iceweasel Aurora

For other builds of Iceweasel, you can get more instructions here.