Puzzler's Edge

Here's an interesting one I stumbled across while doing computer graphics:

How many Eulerian paths are needed to cover all the edges of a cube? (They need not be Eulerian cycles.)

Or, for those unfamiliar with graph theory, if you are walking along the edges of a cube, but you cannot revisit any edges and must restart at a different vertex whenever you have nowhere to go, what is the minimum number of times you have to restart?

Prove your answer.

Bonus points: Generalize your answer to a formula for all hypercubes of an odd dimension. (All even-dimensional ones only require one path)