Microsoft says that the newly announced Xbox One is not backward compatible with the 360:

Xbox One hardware is not compatible with Xbox 360 games. We designed Xbox One to play an entirely new generation of games—games that are architected to take full advantage of state-of-the-art processors and the infinite power of the cloud. We care very much about the investment you have made in Xbox 360 and will continue to support it with a pipeline of new games and new apps well into the future.

This contrasts with the considerable compatibility effort made in the 360, which runs some (but not all) original Xbox games despite having an equally different architecture and a switch from Nvidia to ATI for the GPU. The way this works on the 360 is that when you put in a compatible original Xbox game, it downloads a patch to enable it to run. I am not sure of the details, but there is some kind of compatibility or emulation layer combined with game-specific code to fill any gaps.

This may not seem a big deal to Microsoft, but in a family context it matters. Space in the living room is at a premium in many households, and lack of compatibility means a difficult decision. Replace the old 360 and abandon all that investment in existing games? Have both side by side, adding complexity and clutter? Or pass on the new Xbox and rely on your iPad or Android tablet for fun new games, as the 360 fades from view?

What will happen to classic games as the consoles which run them crumble? Emulation is the answer, and enthusiasts have come up with solutions for many obsolete consoles. In other words, we will end up running those games on PCs. For example, check out Cxbx for an ongoing effort to run original Xbox games, though progress is slow.