Mallard upto griz the .284 & .308 cals are very popular in the west due to bullet offerings, flat shooting, SD and down range energy. Cartridges: 280 rem, 7mm rem mag, 7mm STW, 7mm RUM, etc for the .284 class. .308 cal cartridges: 308win, 3006, 300's (WM, WSM,Wby& RUM). To a lesser extend the .277 cal in 270 win, wby and WSM are also used upto moose sized game.
For grizzly I'd personally feel more comfortable in the .338-.375 cal range.
For deer,sheep, etc.257-.308 are used