Fulfilling the President's Vision for Space Exploration (VSE) will require overcoming many challenges. Among these are the hazards of space radiation to crews traveling to the Moon and Mars. To explore these challenges in some depth and to examine ways to marshal research efforts to address them, NASA, NSF, and the NRC sponsored a workshop bringing together members of the space and planetary science, radiation physics, operations, and exploration engineering communities. The goals of the workshop were to increase understanding of the solar and space physics in the environment of Earth, the Moon, and Mars; to identify compelling relevant research goals; and discuss directions this research should take over the coming decade. This workshop report presents a discussion of radiation risks for the VSE, an assessment of specifying and predicting the space radiation environment, an analysis of operational strategies for space weather support, and a summary and conclusions of the workshop.