This chapter reviews the field of non-linear optics. The practical aspects of non-linear optics are governed by the interaction of electromagnetic radiation with matter; at molecular level this interaction polarizes charge distribution and alters the propagated field. The linear response is described by polarizability and the non-linear response is described by hyperpolarizabilities. The chapter reviews calculations of non-linear-optical properties on small systems such as, He, H2, D2, where quantum chemistry has had a considerable success and to the degree that the results can be used to calibrate experimental equipment. It also deals with the increasing number of papers on ab initio calculations of frequency-dependent first and second hyperpolarizabilities. This is followed by a discussion of the effect that electric fields have on the nuclear, as opposed to the electronic motions in a molecule, and which leads, in turn, to the vibrational hyperpolarizabilities. The chapter also discusses the dispersion formulas that aid in the comparison of hyperpolarizabilities obtained from different processes. © 1994, Academic Press Inc.