The roughly uniform hundredfold depletion of observable Martial atmophiles (nonradiogenic noble gases and reconstituted nitrogen) with respect to Earth implies that Mars lost its atmosphere by a relatively efficient, nonfractionating process. Impact erosion (expulsion of atmosphere by impacts) is an appealing candidate. Noble gases can be used to test this hypothesis. Xenon in particular can be used to impose three constraints on how Mars lost its atmosphere: its very low abundance compared to Earth, Venus, and likely meteoritic sources; its distinctive isotopic composition compared to likely meteoritic sources; and the relatively high absolute abundance of radiogenic 129Xe, daughter of the extinct radionuclide 129I (half-life 17 m.y.). A fourth useful constraint is imposed by radiogenic 40Ar. If produced by impact erosion, the first two constraints become constraints on the composition, mass distribution, and orbital elements of the impactors. The third and fourth constraints imply that Mars lost its nonradiogenic noble gases early, perhaps before it was 100 Myr old. Impact erosion can be invoked to explain Mars by any of three stories: (1) Mars is unlikely. In a sort of planetary brinkmanship, impact erosion almost removed the entire atmosphere but was arrested just in time. (2) Martian noble gases are cometary and cometary Xe is as isotopically mass fractionated as Martian and terrestrial Xe. This is most easily accomplished if a relatively thick geochemically controlled CO2 atmosphere protected trace atmophiles against escape. It is not known if comets actually have the desired composition. (3) Mars was indeed stripped of its early atmosphere but a small remnant was safely stored in the regolith, later released as a by-product of water mobilization. ¿ American Geophysical Union 1993