Large regions of the jovian moon Ganymede have been resurfaced, but the means has been unclear(1,2). Suggestions have ranged from volcanic eruptions of liquid water(3-5) or solid ice(6) to tectonic deformation(7-9), but definitive high-resolution morphological evidence has been lacking. Here we report digital elevation models of parts of the surface of Ganymede, derived from stereo pairs combining data from the Voyager and Galileo spacecraft, which reveal bright, smooth terrains that lie at roughly constant elevations 100 to 1,000 metres below the surrounding rougher terrains. These topographic data, together with new images that show fine-scale embayment and burial of older features(10), indicate that the smooth terrains were formed by flooding of shallow structural troughs by low-viscosity water-ice lavas. The oldest and most deformed areas (the 'reticulate' terrains) in general have the highest relative elevations, whereas units of the most common resurfaced type-the grooved terrain-lie at elevations between those of the smooth and reticulate terrains. Bright terrain, which accounts for some two-thirds of the surface, probably results from a continuum of processes, including crustal rifting, shallow flooding and groove formation. Volcanism plays an integral role in these processes, and is consistent with partial melting of Ganymede's interior(11,12).