A theory of multiphonon relaxation for electronically or vibrationally excited impurities in crystals is developed. Two alternative approaches within the static-coupling scheme are presented and their relative importance is discussed. For each approach a closed-form analytical expression for the relaxation rate is given as a function of energy gap and temperature. We consider relaxation either by optical or acoustic phonons. From the analytical expressions we derive approximate energy gap laws for the zero-temperature rates, and discuss the origin and applicability of a popular phenomenological assumption for the rate’s temperature dependence. The predictions of the theory are compared with energy gap- and temperature-dependent experimental data for various electronic transitions of rare-earth impurities in YA1O(3).