Circumstellar dust exists around several hundred main sequence stars. For the youngest stars, that dust could be a remnant of the protoplanetary disk. Mostly it is inferred to be continuously replenished through collisions between planetesimals in belts analogous to the Solar System's asteroid and Kuiper belts, or in collisions between growing protoplanets. The evolution of a star's debris disk is indicative of the evolution of its planetesimal belts and may be influenced by planet formation processes, which can continue throughout the first gigayear as the planetary system settles to a stable configuration and planets form at large radii. Evidence for that evolution comes from infrared photometry of large numbers of debris disks, providing snapshots of the dust present at different evolutionary phases, as well as from images of debris disk structure. This review describes the theoretical framework within which debris disk evolution takes place and shows how that framework has been constrained by observations.