If ice is present throughout the cryosphere anywhere on Mars, thermodynamic considerations suggest that it is most likely at the poles. Given this condition, the deposition of dust and H2O at the polar surface will ultimately result in a situation where the equilibrium depth to the melting isotherm has been exceeded, melting ice at the base of the cryosphere until thermal equilibrium is once again established. Should deposition persist, the polar deposits will ultimately reach a thickness where melting will occur at their actual base. At this point the cap may reach a state of equilibrium, where the deposition of any additional ice is balanced by geothermal melting. Thermal calculations yield basal melting thicknesses that are consistent with the inferred 4--6 km thickness of the present north polar cap; however, in the south the deposits appear sufficiently thin (1--2 km) that geothermal melting is likely to be relegated to a depth that lies well below the regolith-polar cap interface. Similar conclusions are reached from consideration of the polar caps' theoretical equilibrium profiles. In the north a basal yield stress characteristic of ice at or near the melting point is indicated, while in the south it appears the cap has yet to achieve the necessary height for significant deformation to occur at its base. The potential importance of basal melting is illustrated through the discussion to occur at its base. The potential importance of basal meltin is illustrated through the discussion of four examples: (1) the origin of the major polar reentrants, (2) the removal and storage of an ancient Martian ice sheet, (3) the mass balance of the polar terrains, and (4) the possibility of basal melting at temperate latitudes. This analysis suggests that the process of basal melting may play a key role in understanding the evolution of the Martian polar terrains and the long-term climatic behavior of water on Mars. ¿ American Geophysical Union 1987