An offside offense is punished with an indirect free kick for the opposing team.
Note that the free kick is awarded for the actual offense of offside, not just for being in an offside position--being in an offside position while not participating in play or gaining an advantage is not an offense and doesn't result in any free kick.
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Deliberate handling is a direct free kick offense. However, a goalkeeper who handles the ball within his own penalty area, having received it directly from a teammate's pass, teammate's throw-in, or at any time for longer than 6 seconds, will have committed an indirect-free-kick infraction.