Is the Playstation 4 backwards-compatible?

Does the PS4 have backwards compatibility of some form?

The majority of forums/sites suggest not, or posters are adamant it doesn't yet. Others imply that some form of backwards compatibility will include a cloud service and PS3 titles?

I've even heard on some forums that PS4 does actually play PS3 games but those posts seem to be a minority compared to those that say it's not...

Solution 1:

No, there is no backwards compatibility.

But there are some digital games that will be available on PS4 if you have bought them on PS3/Vita.

The following information can be found in this FAQ

Can I play my PS3 games on my PS4 system?

No, PS4 does not feature backward compatibility for PS3, PS2, or PSone games.

Will any PlayStation Store content purchased on PS3 or PS Vita systems be transferable to a PS4 system?

In some cases, yes. For example, a selection of popular digital titles including Flower, flOw, Escape Plan and Sound Shapes will be available to purchase on PS4 after launch. Gamers who purchased these titles for the PS3 or PS Vita systems (the PS Vita system only, in the case of Escape Plan) will be able to download the PS4 versions of these titles, some of which feature enhanced graphics or improved features, to their PS4 systems for no additional charge.

I heard I can “upgrade” certain PS3 games to work on a PS4 system? How does that work?

Yes, for a limited time and with a $9.99 upgrade fee, you can digitally “upgrade” your purchase of three currently supported games: Call of Duty: Ghosts, Assassin’s Creed IV Black Flag, and Battlefield 4 to its full PS4 version. Click here for further details.

Solution 2:

As far as I know it has no backwards-compatibility but Sony said they will support some kind of backwards compatibility via cloud technology (they bought Gaikai).

References:

• PlayStation 4 will not have native PS3 backwards compatibility
• Yoshida: PSN games won't run natively on PS4, no emulation or cloud support yet
• PlayStation 4 will stream PS1, PS2, PS3 games
• PlayStation 4 Won’t Play PlayStation 3 Games Natively, Coming This Holiday