The ball must be kicked after each score and to start each half (and overtime). PATs (point after touchdown) are mandatory if no 2-point conversion rule is in place. All other kicks are optional.
The football may be punted on any play. A punt before fourth down is usually attempted as a quick kick. (Punts and quick kicks must be from behind the line of scrimmage.) Field goals also may be attempted on any down. PATs may be attempted immediately after a touchdown. A fair catch kick for 3 points may be attempted after a fair catch.
Other than that, you are not allowed to kick the ball.
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In soccer: All players kick the ball.
In Australian football, all players kick the ball.
In the Rugby codes, it is mostly the 'running players' (the 'backs', rather than forwards). Even then, among the backs, more often the Full back or the centres will kick for touch.
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If you are talking about American Football, only the punter and field goal kickers kick the football (unless I am forgetting any...). Usually the punter "kicks off" to start the game, or after a score, or after halftime. The punter also "punts" the ball on 4th down if the team wants to punt, and the field goal kicker kicks field goals when the team wants to.
The quarterback may not advance the ball to another player by kicking it. However, the quarterback may also play the position of punter or place kicker if needed.
yes but only if the ball gets into the hands of the reciving team and he does not go down.