The health risks of space radiation are arguably the most serious challenge to space exploration, possibly preventing these missions due to safety concerns or increasing their costs to amounts beyond what would be acceptable. Radiation in space is substantially different from Earth: high-energy (E) and charge (Z) particles (HZE) provide the main contribution to the equivalent dose in deep space, whereas $\gamma${}~rays and low-energy $\alpha${}~particles are major contributors on Earth. This difference causes a high uncertainty on the estimated radiation health risk (including cancer and noncancer effects), and makes protection extremely difficult. In fact, shielding is very difficult in space: the very high energy of the cosmic rays and the severe mass constraints in spaceflight represent a serious hindrance to effective shielding. Here the physical basis of space radiation protection is described, including the most recent achievements in space radiation transport codes and shielding approaches. Although deterministic and Monte~Carlo transport codes can now describe well the interaction of cosmic rays with matter, more accurate double-differential nuclear cross sections are needed to improve the codes. Energy deposition in biological molecules and related effects should also be developed to achieve accurate risk models for long-term exploratory missions. Passive shielding can be effective for solar particle events; however, it is limited for galactic cosmic rays (GCR). Active shielding would have to overcome challenging technical hurdles to protect against GCR. Thus, improved risk assessment and genetic and biomedical approaches are a more likely solution to GCR radiation protection issues.