Fluviatile and volcanic Martian channels, first discovered on Mariner 9 pictures, have been reexamined by using Viking orbital photography. The superior discrimination of the Viking photography, resulting from clearer atmospheric conditions and an improved camera system, has permitted us to map additional channels and to estimate their relative ages, using a technique based on crater counting. Broad channels like the Ares and Tiu/Simud valles are situated along the margin of the southern highlands near Chryse Planitia, the landing site of Viking 1. They originate in areas of collapsed terrain that may have been formed when subsurface water-ice (permafrost) was melted by geothermal heat from deep-seated volcanic centers. When permafrost melting reached an abrupt topographic slope, the interstitialy stord meltwater 'lakes' were breached suddenly, releasing the great floods that modified the channels. The volume of material involved in the collapsed terrain is large enough to furnish the water calculated to have filled the broad channels. Conditions are reviewed for persistence of liquid water on Mars under present and more favorable pressures and temperatures. Sinuous channels of intermediate size, like the Ma'adim and Hrad valles and other shorter, stubby channels, have multiple tributaries; in the limited coverage available, they appear to result from 'spring sapping,' with the underground permafrost meltwater emerging in box canyons at their heads. The widespread distribution of this type of channel makes their origin by local geothermal heating less likely: climatic warming may be required to explain their formation. The final fluviatile type, dendritic channel networks, has the widest areal distribution and appears to have been formed during at least two episodes. The filamentous channels in their source areas (often the rims of craters) seem to resemble terrestrial river systems; rainfall would seem to be required to form these features. All these channel types debouch onto lowland plains or crater floors, where they disappear in short distances; these abrupt terminations may have resulted from percolation and/or evaporation. Simple and complex lava channels are common; they originate at volcanic centers and are usually morphologically distinct from the aqueous channels. Three types of lava channels are recognized. The wide variation in crater densities implies varying channel ages. Water must have flowed on the Martian surface at many different times in the past, although this would be possible only with great difficulty under the present Martian thermal conditions. Based on a crater flux curve derived by Soderblom et al. (1974) the fluviatile channel ages vary from 3.5 to 0.5 Gy. Lava channel ages range from 3.5 Gy to an age too young to date by the crater counting technique (perhaps 200 m.y.). Methods for dating the channels and volcanic episodes are still insufficiently developed to determine whether episodes of volcanic heating and climatic change are coincident. It is possible that the large floods and volcanic eruptions might trigger a short 'interglacial' interval. Alternatively, the floods may be related to episodic volcanic activity, and the dendritic channels to rainfall that was associated with independent interglacial climatic episodes resulting from variations in solar output or other causes.