This review presents the results of experimental and theoretical studies to detect and investigate deep levels associated with impurities and intrinsic defects in A$^{\mathrm{IV}}$B$^{\mathrm{VI}}$ semiconductors. Group-III impurities are discussed in greatest detail. The experiments (electrophysical, optical, thermophysical) indicate the existence of localized and resonance states in materials doped with indium and thallium, and also (less unambiguously) with gallium and aluminum. Stabilization of the chemical potential is especially clearly manifested in indium doping, and it leads to an extremely high electrical homogeneity of the specimens, as revealed by the long-term relaxation of the concentration of nonequilibrium electrons. The Fermi level substantially varies with the composition of the material, the temperature, and the pressure. Upon doping with thallium, one observes a strong resonance scattering of holes, an electronic heat capacity arising from the impurity, and superconductivity caused by the presence of resonance states. The theoretical and experimental data on localized and resonance states associated with vacancies and complexes of intrinsic defects, and also with impurities of transition metals, bismuth, cadmium, tin, and germanium are reviewed. The genesis of the levels, the energy of interaction of electrons at an impurity center, relaxation mechanisms, and superconductivity are discussed.