Immediately following the formation of the Solar System, small planetary bodies accreted, some of which melted to produce igneous rocks. Over a longer timescale (15-33 Myr), the inner planets grew by incorporation of these smaller objects through collisions. Processes operating on such asteroids strongly influenced the final composition of these planets, including Earth. Currently there is little agreement about the nature of asteroidal igneous activity: proposals range from small-scale melting, to near total fusion and the formation of deep magma oceans. Here we report a study of oxygen isotopes in two basaltic meteorite suites, the HEDs (howardites, eucrites and diogenites, which are thought to sample the asteroid 4 Vesta) and the angrites (from an unidentified asteroidal source). Our results demonstrate that these meteorite suites formed during early, global-scale melting (> or = 50 per cent) events. We show that magma oceans were present on all the differentiated Solar System bodies so far sampled. Magma oceans produced compositionally layered planetesimals; the modification of such bodies before incorporation into larger objects can explain some anomalous planetary features, such as Earth's high Mg/Si ratio.