Recent Viking Orbiter observations of the Martian polar layered deposits, thought to be thick accumulations of dust, have provided new insights into the mechanisms of topographic evolution of these terrains. Here we report the discovery of wavelike topographic relief on north polar surfaces previously supposed to be flat plains. These undulations have wavelengths of a few kilometers and amplitudes of probably only a few ten of meters. The crests and troughs extends laterally up to hundreds of kilometers and have radii of curvature in plan of several hundred kilometers. Much steeper slopes, exhibiting fine terracing due to layering in the deposits, occur within troughs that separate areas of undulating plains and that parallel in strike the topographic grain of the undulations. In the model presented here, both the undulations on the plains surfaces and the layers that appear in cross section on steeper slopes are attributed to the same basic cause--periodic episodes of dust accumulation. These are probably associated with variations in global dust storm intensity and perennial polar ice cap size, brought about by periodic changes in the eccentricity of the Martian orbit and the orientation of its spin axis with respect to the ecliptic. In earlier work it has been proposed that each layer formed in a single cycle of deposition. Specifically, it was argued that dust from global dust storms is swept from the atmosphere by the accumulating annual ice and retained in the place of deposition by a matrix of perennial ice. Here we propose a mechanism for generating the undulating topography at the edge of the advancing or receding perennial ice cap in an environment of periodic deposition of dust. We have recognized several families of topographic undulations with markedly different orientations. In regions where two differently oriented sets of undulations overlap, the wave crests and troughs break up into mounds and shallow basins. We attribute this type of landscape to the occurrence of two passages of the perennial ice cap edge across the same area with the ice cap margin oriented differently on the two occasions. This implies past variations in both the size of the perennial polar cap and the location of its center if we assume that the shape of the perennial cap has not varied drastically from circular. Also reported here is evidence for a genetic relationship between undulating plains and the curvilinear defrosted bands which are the dominant physiographic features of the summer residual polar cap. We establish here that most of these curvilinear features are topographic troughs, and we propose that they are derived from a primary undulating landscape by an insolation-controlled process. First, areas of steeper slopes within the undulating terrain los their perennial ice cover because of a change in insolation conditions. Then, these slopes evolve into topographic troughs because in the absence of perennial ice cover they are unable to trap deposited dust. We attribute the planimetric complexity of the troughs-sharp changes in orientation, lateral offsets and branching-to derivation from intersecting sets of primary topographic undulations with different orientations. In summary, undulating landscapes and trough complexes of the Martian polar layered deposits appear to record both lateral migration and expansion and contraction of perennial ice caps over geological history. Detailed mapping of the planimetric and stratigraphic relationships of the polar landscapes may ultimately provide the basis for a detailed model of landscape evolution and a historical record of variations on the size and location of the perennial ice cap.