Many solar system objects, such as planets, satellites, rings, and comets, are known to either be made of ices or to have icy surfaces. Because these ices are subjected to radiation processing by keV-MeV ions and UV photons, their chemical and physical properties evolve over time. We have employed a laboratory approach to investigate these radiolytic and photolytic changes. Ices, either pure or mixtures, are subjected to ionizing radiation and UV photons and then probed by IR spectroscopy. It has been found that radiolysis and photolysis destroy reactant molecules, synthesize new molecules, cause changes of phase in pure materials, and eject molecules from ices. Although our laboratory approach initially focused on product identification, it has become increasingly necessary to work toward a comprehensive understanding of ice chemistry. To this end, we describe recent radiation experiments on some of the most important solar system and interstellar ice molecules, such as H2O, CO, CO2, and hydrocarbons. Chemical reactions are given to account for our observations, and these reactions include such general types as acid-base, electron-transfer, radical-radical combination, and H-atom addition. Comparisons of ion and UV processing are described for pure H2O and for two binary mixtures, H2O + CO2 and NH3 + CO. Predictions of molecular evolution in complex ices, both H2O- and N2-dominated, are made. Applications are described for cometary organic chemistry, ion processing in interstellar ices, and the recent discovery of H2O2 on the surface of Europa.