The interstellar medium is characterized by a rich and diverse chemistry. Many of its complex organic molecules are proposed to form through radical chemistry in icy grain mantles. Radicals form readily when interstellar ices (composed of water and other volatiles) are exposed to UV photons and other sources of dissociative radiation, and if sufficiently mobile the radicals can react to form larger, more complex molecules. The resulting complex organic molecules (COMs) accompany star and planet formation and may eventually seed the origins of life on nascent planets. Experiments of increasing sophistication have demonstrated that known interstellar COMs as well as the prebiotically interesting amino acids can form through ice photochemistry. We review these experiments and discuss the qualitative and quantitative kinetic and mechanistic constraints they have provided. We finally compare the effects of UV radiation with those of three other potential sources of radical production and chemistry in interstellar ices: electrons, ions, and X-rays.