Adobe Air has been released for Linu, ALPHA-quality. See how to install it and launch Applications.
You can download the Adobe Runtime here. It is a bin file. Remember where you download the file (it is called something like adobeair_linux_a198389.bin). make it executable (chmod +x adobeair_linux_a1_*****.bin) and launch it (./adobeair_linux_*****.bin).
Agree to the terms and let the installation finish. Of course you have the runtime environment installed now. What you need is an application to run. Go here , to download a sample application.
Just click the sample application and Firefox opens a dialog to open the Application with Adobe AIR. Just click OK. And AIR is launched to install the sample Application you have chosen. To launch an Application double-Click the icon on the DESKTOP, or if not present select your application from the Start Menu -> Adobe AIR -> Your App.
To download the developer SDK for Linux, go here. And download the source code of an application here. Extract both archives where you want.(I got a warning that the source code of the application is not to be trusted and might present a security risk, but for you out there I engaged this real dangerous treat!:))
Now to launch the application in debug mode, change directory (cd) to the adobeair_linux_sdk/bin. And then launch the application in debugger mode by running ./adl /and/path/to/your/application.xml file.
To package your application for distribution find all necessary informations here.