In football, the ball starts at the line of scrimmage and from there is hiked back to the quarterback. Doing this creates room for the quarterback to make passing or running plays. Clearly hiking the ball is a good thing; but do the rules absolutely require it? Or, as bad an idea as it may be, could a team try to move the ball forwards directly from the line of scrimmage with out hiking it back first?
In the NFL the snap must be taken by someone who is not on the line of scrimmage - so none of the 5 ineligible receivers can be the first to handle a snap (guard grabbing snap for instance).
The only caveat to this is anyone can touch the ball once it has hit the ground. There are several trick plays usually ran in inclement weather (that hides ball view) where center snaps ball to ground and guard scoops and runs.
The NCAA has similar rules/restrictions. However when you get to high school and below depending on the state/league the restrictions on the hand-offs can go up. For instance in a league I coached in I ran a play where the guard received the ball while pulling. The next year it was banned. In fact in almost all pee-wee leagues there are usually rules enforcing that if a lineman has the ball it is dead.
Why doesn't the center (or guard) run the ball in the NFL/NCAA? Two reasons - First if you have a guy that gets blown up before receiving the ball then that is high risk for turnover - remember you will be putting the lineman in a vulnerable position. The second is the reward. There is little reward in giving the ball to a 300 pound lineman who is barely moving.