River-crossing classics

For this week's puzzle, we will have two simple classics. Remember: in the world of puzzles, simple is not necessarily bad, and challenging is not necessarily good.

1. A man must cross the river with a fox, a chicken, and some cabbage. There is only one boat, and only the man can row it, and he can only take one companion at a time. For obvious reasons, he cannot leave the fox alone with the chicken or leave the chicken alone with the cabbage. How can he do it?

2. Three missionaries and three cannibals must cross a river (geez, whoever wrote this one had a dark sense of humor). They have a single boat that can only carry two people, and there must be at least one person rowing it. If at any time, there are more cannibals than missionaries on either side of the river, the cannibals will eat the missionaries. How can everyone get across alive?

Update: Solution has been posted