I only kept one PS2 game, Smashed, but it does play (Star Wars Battlefront II). As far as I know the backwards compatibility is already in the software somewhere so there is nothing you have to download. Though I heard that system update 1.5 had some fixes in it. I know I'm way ahead of that update.
From what I know, most games will play but some will not. I think you can find out more at
http://www.us.playstation.com/Support/CompatibleStatus
thre's probably one for Europe, too.
Also found this site:
http://www.ps3comp.com/view.asp
but it seems difficult to decipher as I could not see a legend.