The conditions under which large volumes of water may have flowed at high speeds across the surface of Mars are considered. To assess the likely ranges of initial water temperature and release rate, the possible conditions in subsurface aquifers confined beneath the cryosphere are explored. Then the transfer of water to the surface in fractures induced by volcanic activity or tectonic events is modelled and the physical processes involved in its release into the Martian environment are discussed. The motion of the water across the surface is analysed with standard treatments for fluvial systems on Earth, modified for Mars by taking account of the differing environmental conditions and removing what may be considered to be the unsafe assumption that most channels involved bankfull flows. The most commonly discussed environmental difference is the smaller acceleration due to gravity on Mars. However, an important additional factor may have been the initially vigorous evaporation of water into the low-pressure Martian atmosphere. This process, together with the thermal losses incurred by assimilation of very cold rock and ice eroded from the cryosphere over which the water travels, causes minor changes in the depth and speed of a water flood but, eventually, produces major changes in the flood rheology as the total ice and sediment loads increase. The roles of these processes in determining the maximum distance to which the water may travel, and the relative importance of erosion and deposition in its bed, are discussed.