Abstract River channels and valleys have been observed on several planetary bodies in addition to the Earth. Long sinuous valleys on Venus, our Moon and Jupiter's moon Io are clearly formed by lava, and branching valleys on Saturn's moon Titan may be forming today by rivers of methane. But by far the most dissected body in our Solar System apart from the Earth is Mars. Branching valleys that in plan resemble terrestrial river valleys are common throughout the most ancient landscapes preserved on the planet. Accompanying the valleys are the remains of other indicators of erosion and deposition, such as deltas, alluvial fans and lake beds. There is little reason to doubt that water was the erosive agent and that early in Mars' history, climatic conditions were very different from the present cold conditions and such that, at least episodically, water could flow across the surface. In addition to the branching valley networks, there are large flood features, termed outflow channels. These are similar to, but dwarf, the largest terrestrial flood channels. The consensus is that these channels were also cut by water although there are other possibilities. The outflow channels mostly postdate the valley networks, although most are still very ancient. They appear to have formed at a time when surface conditions were similar to those that prevail today. There is evidence that glacial activity has modified some of the water-worn valleys, particularly in the 30–50° latitude belts, and ice may also be implicated in the formation of geologically recent, seemingly water-worn gullies on steep slopes. Mars also has had a long volcanic history, and long, sinuous lava channels similar to those on the Moon and Venus are common on and around the large volcanoes. These will not, however, be discussed further; the emphasis here is on the effects of running water on the evolution of the surface.