The amount of solar irradiance reaching the surface is a key parameter in the hydrological and energy cycles of the Earth's climate. We analyze 20th Century simulations using nine state-of-the-art climate models and show that all models estimate a global annual mean reduction in downward surface solar radiation of 1--4 W/m2 at the same time that the globe warms by 0.4--0.70C. In single forcing simulations using the GISS-ER model, this "global dimming" signal is shown to be predominantly related to aerosol effects. In the global mean sense the surface adjusts to changes in downward solar flux instantaneously by reducing the upward fluxes of longwave, latent and sensible heat. Adding increased greenhouse gas forcing traps outgoing longwave radiation in the atmosphere and surface which results in net heating (although reduced) that is consistent with global warming over the 20th Century. Over the 1984--2000 period, individual model simulations show widely disparate results, mostly related to cloud changes associated with tropical Pacific variations, similar to the changes inferred from the satellite data analysis. This suggests that this time period is not sufficient to determine longer term trends.