An overview is presented of the theory of X-ray Raman scattering. Second-order perturbation theory for the interaction between matter and light is used as a common starting point, and the consequences of this theory are analytically and numerically analyzed for a variety of experimental situations. The review focuses on results frost radiative and nonradiative scattering experiments conducted with 2nd and 3rd generation synchrotron radiation sources during the last couple of years, dealing with atomic, molecular, solid state and surface adsorbate targets. After giving a brief synopsis of relevant experimental techniques, some basic theoretical concepts and principles of X-ray Raman scattering are described, followed by a presentation of the various particular aspects associated with the resonant X-ray scattering process. That is: polarization interference - role of symmetry - symmetry breaking and energy dependence - dissociation and time dependent interpretations - duration time and frequency detuning - formation of band profiles - Doppler effects - screening and chemical shifts - elastic scattering - solid state theory - application to surface adsorbates - absorption in the Raman mode - direct processes versus resonant X-ray scattering - many channel theory. Each aspect is described by a qualitative picture, a mathematical analysis, and by illustrative examples from experiment combined in some cases with results from simulations. Simple systems are chosen to demonstrate the consequences of various aspects of the theory. (C) 1999 Elsevier Science B.V. All rights reserved.