A key pacemaker of ice ages on the Earth is climatic forcing due to variations in planetary orbital parameters. Recent Mars exploration has revealed dusty, water-ice-rich mantling deposits that are layered, metres thick and latitude dependent, occurring in both hemispheres from mid-latitudes to the poles. Here we show evidence that these deposits formed during a geologically recent ice age that occurred from about 2.1 to 0.4 Myr ago. The deposits were emplaced symmetrically down to latitudes of ~30°-equivalent to Saudi Arabia and the southern United States on the Earth-in response to the changing stability of water ice and dust during variations in obliquity (the angle between Mars' pole of rotation and the ecliptic plane) reaching 30-35°. Mars is at present in an `interglacial' period, and the ice-rich deposits are undergoing reworking, degradation and retreat in response to the current instability of near-surface ice. Unlike the Earth, martian ice ages are characterized by warmer polar climates and enhanced equatorward transport of atmospheric water and dust to produce widespread smooth deposits down to mid-latitudes.