I've fixed all types of stripped and rusted/busted screws and stuff with a drill bit that is designed for it. They're cylindrical and have ridges on them, used to "catch" the sides of a problem screw. There are kits of these, with different size bits and can be found at any home store.
First, you drill into the bolt/screw. The size of drill bit you use depends on the size of the screw/bolt. Then you take one of the special bits and hammer it gently into the hole you've drilled. Again, the size of the bit will depend on the size of the hole you've drilled. You will se a bit that is just a bit larger than the hole you've drilled.
The bit should "stick" inside the bolt. You then fasten your drill onto the bit, select the reverse drive, and slowly work it out. If it doesn't work the first time, just tap the bit in again and try it again.