After having been an instrument of the Cold War, space exploration has become a major technological, scientific and societal challenge for a number of countries. With new projects to return to the Moon and go to Mars, radiobiologists have been called upon to better assess the risks linked to exposure to radiation emitted from space (IRS), one of the major hazards for astronauts. To this aim, a major task is to identify the specificities of the different sources of IRS that concern astronauts. By considering the probabilities of the impact of IRS against spacecraft shielding, three conclusions can be drawn: (1) The impacts of heavy ions are rare and their contribution to radiation dose may be low during low Earth orbit; (2) secondary particles, including neutrons emitted at low energy from the spacecraft shielding, may be common in deep space and may preferentially target surface tissues such as the eyes and skin; (3) a “bath of radiation” composed of residual rays and fast neutrons inside the spacecraft may present a concern for deep tissues such as bones and the cardiovascular system. Hence, skin melanoma, cataracts, loss of bone mass, and aging of the cardiovascular system are possible, dependent on the dose, dose-rate, and individual factors. This suggests that both radiosusceptibility and radiodegeneration may be concerns related to space exploration. In addition, in the particular case of extreme solar events, radiosensitivity reactions—such as those observed in acute radiation syndrome—may occur and affect blood composition, gastrointestinal and neurologic systems. This review summarizes the specificities of space radiobiology and opens the debate as regards refinements of current radiation protection concepts that will be useful for the better estimation of risks.