There are (at least) three separate kinds of answers to this. One is a special mode for the flash, in which it produces a flash (or series of flashes) that last long enough to provide coverage as the "slit" in the shutter moves across the focal plane. It's mostly related to the flash rather than the camera though. AFAIK, Minolta was the first to introduce this; Sony continues to provide it since taking over Minolta's camera business. I believe Canon and Nikon now offer similar capabilities as well.
The second is simply shutters that move the shutter curtain faster to provide a higher sync with a normal flash. I believe there are really only two cameras with such shutters: the Minolta Dynax/Maxxum/Alpha 9xi and the Minolta Dynax/Maxxum/Alpha 9. Both of these provided X-sync at 1/350th of a second. They're both film cameras.
Finally, there are some medium format cameras (e.g., Hassleblads) that accept lenses with a leaf shutter. Unlike a focal plane shutter, a leaf shutter normally provides X-sync right up the top speed of which it's capable -- usually 1/500th of a second.