The theory of the chemical interaction of molecules with surfaces has advanced handsomely in the last few years. This is due in part to the application of the entire arsenal of bulk solid-state theory and molecular quantum chemistry methods. This considerable activity was stimulated by an outpouring of experimental data, particularly of photoemission spectra. In many cases the theoretical techniques are now such that accurate, atomistic pictures of chemisorption phenomena are computed from first principles. This level of capability has been reached only recently, and has not been described anywhere in a comprehensive manner. The purpose of this monograph is to review these recent advances and, at the same time, to indicate a number of important questions which have not been answered. We discuss chemisorption on oxides, semiconductors, and both simple and transition metals. Solid surfaces as well as clusters are considered. While the review should be valuable to workers in the field, care has been taken to make the chapters understandable to the nonspecialist.