This review analyzes the status of the problem of H$^-$-like centers and the molecular complexes that they form in doped semiconductors. The study of these centers is closely associated with the general problem of the localization of electrons in disordered systems. The existing experimental data are discussed from a unitary standpoint. Models of various impurity complexes are discussed and the pertinent estimates are given. We show that all the fundamental observed regularities at low and moderate impurity concentrations are explained by the formation of either isolated H$^-$-like centers or of complexes of the H$^-$--H$^+$-type. At higher impurity concentrations, the number of H$^-$--H$^+$-type complexes declines, impurity clusters are formed, and then a conducting impurity band arises from the H$^-$-like states.