In this introductory chapter, we describe the basic features of diluted magnetic semiconductors. We focus on giant Zeeman splitting of excitons and related giant Faraday rotation. We show that the spin splitting is proportional to the magnetization and develop a simple model, making use of virtual crystal and mean field approximations, describing the energy states in the presence of magnetic field in diluted magnetic semiconductors having moderate and narrow gaps between the valence and conduction bands. We discuss limitations of this model description and mention cases, which make a more refined theories necessary. We introduce and discuss sp − d exchange interaction constants, show the trends existing in their values in the family of II–VI DMSs. We finally discuss the d − d exchange interaction, which is responsible for magnetic properties of the materials in question and show how to conveniently parametrize the magnetization. sp − dd − d