Abstract In Chapter 2 we utilized the basic relation between the double differential scattering cross-section and excitations in the scattering system, as given in equation (2.3) only by considering excitations of the valence electron systems of condensed matter. We will now examine inelastic scattering processes connected with excitations of core-electrons from closed shells into unoccupied states about which we want to learn by this type of spectroscopy. In this case we will speak, for more historical reasons, of non-resonant X-ray Raman scattering or spectroscopy, or still more briefly of X-ray Raman scattering or spectroscopy (XRS), if mixing up with resonant X-ray Raman scattering is not to be suspected.