This article reviews the recent discovery and several important applications of recoilless gamma-ray fluorescence. Its main purpose is to give to nonspecialists a self-contained introduction to this currently very lively field of research. The importance of the fluorescence technique and its limitations, before Mössbauer's discovery, are discussed. Mössbauer's pioneering experiments are briefly described and Lipkin's simple theoretical explanation of recoilless gamma-ray emission is presented. From among the large number of recent experiments which make use of the Mössbauer effect, two groups are selected for discussion. The first is concerned with the discovery of nuclear and magnetic properties of iron-57; the other with a test of the principle of equivalence in general relativity, by a measurement of the gravitational shift in the frequency of radiation.