Space programmes are shifting towards planetary exploration and, in particular, towards missions by human beings to the Moon and to Mars. Radiation is considered to be one of the major hazards for personnel in space and has emerged as the most critical issue to be resolved for long-term missions both orbital and interplanetary. The two cosmic sources of radiation that could impact a mission outside the Earth's magnetic field are solar particle events (SPE) and galactic cosmic rays (GCR). Exposure to the types of ionizing radiation encountered during space travel may cause a number of health-related problems, but the primary concern is related to the increased risk of cancer induction in astronauts. Predictions of cancer risk and acceptable radiation exposure in space are extrapolated from minimal data and are subject to many uncertainties. The paper describes present-day estimates of equivalent doses from GCR and solar cosmic radiation behind various shields and radiation risks for astronauts on a mission to Mars.