Recently, using data from the Chinese lunar probes Chang’e-1 and Chang’e-2, scientists found excessive heat below the surface of a suspected volcanic feature on the Moon known as Compton-Belkovich. The existence of such a large hotspot on the lunar surface can only be explained if the underlying granite rock contains elements undergoing radioactive decay, such as thorium and uranium.
Researchers estimate that the level of this background radioactivity is about 0.3 millisieverts per year. This is actually small compared to the approximately 6 millisieverts per year that every person on Earth receives annually, due to natural sources of radiation.
There are however other sources of radiation that being on the Moon would expose you to, which you are protected from on Earth due to the planet’s magnetic field. These include galactic cosmic rays, energetic particles from the Sun, and neutrons and gamma rays from interactions between radiation and the lunar soil.