The increase in solar luminosity over geological timescales should warm the Earth’s climate, increasing water evaporation, which will in turn enhance the atmospheric greenhouse effect. Above a certain critical insolation, this destabilizing greenhouse feedback can ‘run away’ until the oceans have completely evaporated. Through increases in stratospheric humidity, warming may also cause evaporative loss of the oceans to space before the runaway greenhouse state occurs. The critical insolation thresholds for these processes, however, remain uncertain because they have so far been evaluated using one-dimensional models that cannot account for the dynamical and cloud feedback effects that are key stabilizing features of the Earth’s climate. Here we use a three-dimensional global climate model to show that the insolation threshold for the runaway greenhouse state to occur is about 375 W m−2, which is significantly higher than previously thought. Our model is specifically developed to quantify the climate response of Earth-like planets to increased insolation in hot and extremely moist atmospheres. In contrast with previous studies, we find that clouds have a destabilizing feedback effect on the long-term warming. However, subsident, unsaturated regions created by the Hadley circulation have a stabilizing effect that is strong enough to shift the runaway greenhouse limit to higher values of insolation than are inferred from one-dimensional models. Furthermore, because of wavelength-dependent radiative effects, the stratosphere remains sufficiently cold and dry to hamper the escape of atmospheric water, even at large fluxes. This has strong implications for the possibility of liquid water existing on Venus early in its history, and extends the size of the habitable zone around other stars.