Black bear you can use anything that works solidly on deer - minimum 6mm caliber I'd say.
The great bears, I would say my personal minimum calibers are .35 Whelen or 9.3x62 in a bolt and .348 Winchester in a lever gun. .375 H&H is preferable.
Personally though I like to over-gun in terms of sectional density for black bear. They have fat that tends to seal up their wounds, so you don't get much blood for tracking if necessary from the entrance wound. That means I darn well want an exit wound. I've had good luck with a 7mag and 175gr A-frames. Woodleigh makes a 175gr 7mm protected point that should be suitable for the same reason although I've never actually used it.
