Ok, so I've been muttering a lot recently about Facebook and Flash. It's something I'm interested in - because I believe it's one great way to expose your Flash games to lots of users - but just not something I've had time to bring to fruition.
It turns out your basic requirements for integrating Flash and Facebook boil down to this - a server-side solution (asp, .Net, PHP, Ruby etc.) for building webservices, or some form of data service for Flash to work with. Most of the integration actually, will be done with server-side stuff. And then you just need to build a pretty standard Flash application that uses these webservices.
Beyond that fantastically helpful statement, here's a couple of links to help you on your way;