We review our past and ongoing use of ground-penetrating radar to investigate permafrost in Alaska and in the Dry Valleys of Antarctica. The results may be relevant to radar efforts on Mars because of arid conditions and the presence of ice. The pulses were centered at 50, 100, and 400 MHz. We interpret profiles from two sites in the eastern Taylor Valley to show glaciolacustrine and glaciofluvial stratigraphy. The maximum depth of stratigraphy profiled there was about 33 m. Near Fairbanks, Alaska, the depth of penetration at 50 MHz was near 80 m in marginally frozen and stratified alluvial sands. At the Fairbanks sites, supplementary drilling was required to differentiate between reflections from conductive bedrock, a graphitic schist, and those from the water table at depths of 20--25 m. At a site on the North Slope of Alaska, we profiled present and remnant freezing fronts in an alluvial floodplain. The relative permittivity at most sites ranged between about 4 and 5.5, which is consistent with dry conditions, the mineralogy, and low ice content. Weak interface reflectivity or the lack of further interfaces may have limited the interpretation of maximum penetration where no water table was present because signal absorption should have been low and scarce diffractions imply that scattering was weak. The interface reflectivities beneath Taylor Valley may be a function of only density contrasts, since free water, and possibly ice, is absent.