The YNAB desktop software was written on the Adobe AIR™ platform starting back in 2009 with our previous version YNAB 3. Using AIR allowed us to maintain a single code base for Windows, Mac and Linux computers. Unfortunately, in June of 2011, Adobe announced they were ceasing support for AIR on Linux leaving us unable to directly support Linux users.
Fortunately, some Linux YNAB users have had success getting the Windows version of YNAB running on their systems using Wine™. While we aren't able to help with the direct troubleshooting due to the various Linux distros you will find an active knowledgable community of Linux users in this thread on our forums ready and willing to help.
Once installed and running YNAB will function exactly the same as its Windows and Mac counterparts with all features intact.