This article describes the application of activation analysis to gold, silver, and copper coins. Irradiations were carried out at the end of the extracted beam of the Instituut voor Kernphysisch Onderzoek synchrocyclotron. Copper and silver in gold coins were determined by 30 MeV proton activation, gold and lead in silver coins by 26 MeV deuteron activation, and iron, nickel, zinc, arsenic, antimony, tin, and lead in copper coins by fast neutron activation. The principal nuclear reactions involved are specified in tables, and typical gamma ray spectra are presented for each type of coin and activation. Formulas for calculating the weight fraction of the elements of interest are presented. Composition of Roman gold coins remained very well-controlled up to about 200 A.D., after which copper and silver content increased slightly. A number of problems of authenticity are discussed. Copper coins were found to be particularly variable in composition. The activation techniques used are argued to have advantages over X-ray fluorescence and thermal neutron activation analysis for the study of coins.