PlayStation 4 PlayStation 3 games
Unfortunately, Sony just announced that the PlayStation 4 won't play PS3 games natively out-of-the-box — all of the games you've accumulated over the last few years will be stuck on the older console. However, Sony did say it is exploring "advanced technologies" to bring PS3 support to the PS4 eventually. That backwards compatibility could come via Gaikai-powered streaming, just as we heard last week. According to a Sony tweet, it sounds like that compatibility won't be limited to just the PS3 — original PS and PS2 games will eventually be available as well. However, at launch, it sounds like you'll be limited to new games only.
Update: Sony's official UK Twitter account has confirmed that legacy games will be playable at some point.