Ionizing radiation and microgravity are two considerable health risks encountered during deep space exploration. Both have deleterious effects on the human body. On one hand, weightlessness is known to induce a weakening of the immune system, delayed wound healing and musculoskeletal, cardiovascular, and sensorimotor deconditioning. On the other hand, radiation exposure can lead to long-term health effects such as cancer and cataracts as well as have an adverse effect on the central nervous and cardiovascular systems. Ionizing radiation originates from three main sources in space: galactic cosmic radiation, solar particle events and solar winds. Furthermore, inside the spacecraft and inside certain space habitats on Lunar and Martian surfaces, the crew is exposed to intravehicular radiation, which arises from nuclear reactions between space radiation and matter. Besides the approaches already in use, such as radiation shielding materials (such as aluminium, water or polyethylene), alternative shielding materials (including boron nanotubes, complex hybrids, composite hybrid materials, and regolith) and active shielding (using fields to deflect radiation particles) are being investigated for their abilities to mitigate the effects of ionizing radiation. From a biological point of view, it can be predicted that exposure to ionizing radiation during missions beyond Low Earth Orbit (LEO) will affect the human body in undesirable ways, e.g., increasing the risks of cataracts, cardiovascular and central nervous system diseases, carcinogenesis, as well as accelerated ageing. Therefore, it is necessary to assess the risks related to deep space exploration and to develop mitigation strategies to reduce these risks to a tolerable level. By using biomarkers for radiation sensitivity, space agencies are developing extensive personalised medical examination programmes to determine an astronaut's vulnerability to radiation. Moreover, researchers are developing pharmacological solutions (e.g., radioprotectors and radiomitigators) to proactively or reactively protect astronauts during deep space exploration. Finally, research is necessary to develop more effective countermeasures for use in future human space missions, which can also lead to improvements to medical care on Earth. This review will discuss the risks space travel beyond LEO poses to astronauts, methods to monitor astronauts' health, and possible approaches to mitigate these risks.