We review the physical properties of diluted magnetic semiconductors (DMS) of the type AII1−xMnxBVI(e.g., Cd1−xMnxSe, Hg1−xMnxTe). Crystallographic properties are discussed first, with emphasis on thecommonstructural features which these materials have as a result of tetrahedral bonding. We then describe the band structure of the AII1−xMnxBVIalloys in the absence of an external magnetic field, stressing the close relationship of thespelectron bands in these materials to the band structure of the nonmagnetic AIIBVI‘‘parent’’ semiconductors. In addition, the characteristics of the narrow (nearly localized) band arising from the half‐filled Mn 3d5shells are described, along with their profound effect on the optical properties of DMS. We then describe our present understanding of themagneticproperties of the AII1−xMnxBVIalloys. In particular, we discuss the mechanism of the Mn++‐Mn++exchange, which underlies the magnetism of these materials; we present an analytic formulation for the magnetic susceptibility of DMS in the paramagnetic range; we describe a somewhat empirical picture of the spin‐glasslike freezing in the AII1−xMnxBVIalloys, and its relationship to the short range antiferromagnetic order revealed by neutron scattering; and we point out some not yet fully understood questions concerning spin dynamics inDMS revealed by electron paramagnetic resonance. We then discuss thesp‐dexchange interaction between thespband electrons of the AII1−xMnxBVIalloy and the 3d5electrons associated with the Mn atoms. Here we present a general formulation of the exchange problem, followed by the most representative examples of its physical consequences, such as the giant Faraday rotation, the magnetic‐field‐induced metal‐to‐insulator transition in DMS, and the properties of the bound magnetic polaron. Next, we give considerable attention to the extremely exciting physics of quantum wells and superlattices involving DMS. We begin by describing the properties of the two‐dimensional gas existing at a DMS interface. We then briefly describe the current status of the AII1−xMnxBVIlayers and superlattices (systems already successfully grown; methods of preparation; and basic nonmagnetic properties of the layered structures). We then describe new features observed in the magnetic behavior of the quasi‐two‐dimensional ultrathin DMS layers; and we discuss the exciting possibilities which thesp‐dexchange interaction offers in the quantum‐well situation. Finally, we list a number of topics which involve DMS but which have not been explicitly covered in this review such as elastic properties of DMS, DMS‐based devices, and the emerging work on diluted magnetic semiconductors other than the AII1−xMnxBVIalloys—and we provide relevant literature references to these omitted topics.