Right now Adobe AIR is only officially available for 32-bit Linux.
But it does work on 64-bit Linux with the 32-bit compatibility libraries.
There are several ways to install Adobe AIR on Linux.

My preferred way on Ubuntu is to use the .deb package.
However the .deb package distributed by Adobe can only be installed on 32-bit systems.

To install the Adobe AIR .deb package on a 64-bit system just follow these steps:

1. Download the Adobe AIR .deb file
2. In a terminal window go to the directory containing the adobeair.deb file
3. Create a tmp dir:

   mkdir tmp

4. Extract the deb file to the tmp dir:

   dpkg-deb -x adobeair.deb tmp

5. Extract the control files:

   dpkg-deb --control adobeair.deb tmp/DEBIAN

6. Change the Architecture parameter from “i386″ to “all”:

   sed -i "s/i386/all/" tmp/DEBIAN/control

7. Repackage the deb file:

   dpkg -b tmp adobeair_64.deb

   dpkg-deb: building package `adobeair' in `adobeair_64.deb'.

Now you can install Adobe AIR on a 64-bit system! From the command line just do:

sudo dpkg -i adobeair_64.deb

Selecting previously deselected package adobeair.(Reading database ... 158232 files and directories currently installed.)
Unpacking adobeair (from adobeair_64.deb) ...

Running 32-bit AIR on 64-bit systems has not been fully tested. However, 32-bit AIR is expected to run on 64-bit systems if they have the required 32-bit libraries and packages installed.
The following links describe how to install or update AIR 2.6 on 64-bit versions of Ubuntu 9.10, Fedora Core 12 and OpenSuse 11.2.
Installing Adobe AIR 2.6 on 64-bit Linux visit http://kb2.adobe.com/cps/521/cpsid_52132.html
Updating to Adobe AIR 2.6 on 64-bit Linux visit http://kb2.adobe.com/cps/521/cpsid_52132.html
Setting up adobeair (1:2.6.0.19120) ...