Human exploration of the solar system beyond Earth's orbit will entail many risks for the crew on these deep space missions. One of the most significant health risks is exposure to the harsh space radiation environment beyond the protection provided by the Earth's intrinsic magnetic field. Crew on exploration missions will be exposed to a complex mixture of very energetic particles. Chronic exposures to the ever-present background galactic cosmic ray (GCR) spectrum consisting of all naturally occurring chemical elements are combined with sporadic, possibly acute exposures to large fluxes of solar energetic particles, mainly protons and alpha particles. The background GCR environment is mainly a matter of concern for stochastic effects, such as the induction of cancer with subsequent mortality in many cases, and late deterministic effects, such as cataracts and possible damage to the central nervous system. Unfortunately, the actual risks of cancer induction and mortality owing to the very important high-energy heavy ion component of the GCR spectrum are essentially unknown. The sporadic occurrence of extremely large solar energetic particle events (SPE), usually associated with intense solar activity, is also a major concern for the possible manifestation of acute effects from the accompanying high doses of such radiations, especially acute radiation syndrome effects such as nausea, emesis, haemorrhaging or, possibly, even death. In this presentation, an overview of the space radiation environment, estimates of the associated body organ doses and equivalent doses and the potential biological effects on crew in deep space are presented. Possible methods of mitigating these radiations, thereby reducing the associated risks to crew are also described.