Grazing impacts can be identified on the basis of the elongate shape of the resulting crater and a distinctive pattern of ejecta deposits. Over 170 such impact craters larger than 3 km are recognized on Mars, and they represent more than 5% of the total crater population of the ridged plains. In contrast, the moon exhibits only one comparable example larger than 3 km on the maria, a frequency consistent with theoretical estimates for an isotropic influx of impactors. Many Mars grazers appear to occur along great circles. The most recent examples generally impacted in an east-west direction, whereas older grazers impacted in more northerly directions. We interpret the excessive number of grazers and the common impact directions as the result of satellites whose orbits tidally decayed with time. If all orbits originally had small inclinations similar to the orbits of Phobos and Deimos as well as the most recent grazers, then the change in impact direction with time can be explained as the result of shifts in the crust due to changes in the martian moments of inertia. The locations of the projected orbital axes (orbit-pole points) on the martian surface indicate that the geographic poles of Mars originally were situated at lower latitudes. More than 95% of the mass represented by these proposed satellites impacted prior to the emplacement of the volcanic plains of Lunae Planum. The estimated combined mass of grazing impactors would form a satellite at least 225 km in diameter. These results may provide new clues for the origin of Phobos and Deimos and perhaps the angular momentum of Mars.