This chapter is devoted to the quantum formulation of the phenomenon of x-ray dichroism, i.e. the polarization dependence of resonant core to valence transitions. In quantum theory, the four types of polarization dependent absorption effects, XNLD, XMLD, XNCD, and XMCD, whose historical development has been outlined in Sect. I.6.9 (see Fig. I.6.19), arise from non-spherically symmetric charge or spin densities in the atomic volume. Owing to the tunability of photon energy and polarization at modern x-ray sources, dichroic studies have evolved into important and in many cases unique applications of x-rays. Here we treat this topic with emphasis on fundamental concepts that are illustrated by experimental results chosen to complement those in previous books [1–3] and the review by Wende [4].