This paper contains two main sections. In the first one, we illustrate the technological interest of crystals of the sillenite family and we present our powerful methods of investigation in some details. These are essentially optical absorption and magnetic circular dichroism at liquid helium temperatures, down to 4000 cm-1. The latter is also used to detect electron paramagnetic resonance optically and to tag a given absorption component to a specific paramagnetic, intrinsic or extrinsic, defect. In the second part, we present a variety of experimental results and interpretations dealing with undoped crystals and with samples doped with transition ions of the iron (Mn, Cr, Fe, Co, Cu) group. Emphasis is put on the assignment of ligand field bands in the near infrared, since these are used to monitor the site, charge state, and amount of the dopants in a thermally bleached state or after various illuminations.