A one-dimensional climate model is used to study the response of an Earth-like atmosphere to large increases in solar flux. For fully saturated, cloud-free conditions, the critical solar flux at which a runaway greenhouse occurs, that is, the oceans evaporate entirely, is found to be 1.4 times the present flux at Earth's orbit (S0). This value is close to the flux expected at Venus' orbit early in solar system history. Is is nearly independent of the amount of CO2 present in the atmosphere, but is sensitive to the H2O absorption coefficient in the 8- to 12-micrometers window region. Clouds should tend to depress the surface temperature on a warm, moist planet; thus, Venus may originally have had oceans if its initial water endowment was close to that of Earth. It lost them early in its history, however, because of rapid photodissociation of water vapor followed by escape of hydrogen to space. The critical solar flux above which water is rapidly lost could be as low as 1.1S0. The surface temperature of a runaway greenhouse atmosphere containing a full ocean's worth of water would have been in excess of 1500 degrees K--above the solidus for silicate rocks. The presence of such a steam atmosphere during accretion may have significantly influenced the early thermal evolution of both Earth and Venus.