The good solubility of Mn atoms in a matrix based on gapless semiconductors HgTe, HgSe makes it possible to reconstruct smoothly the band structure of solid solutions Hg$_{1–x}$Mn$_x$Te and Hg$_{1–x}$Mn$_x$Se from the inverted gapless structure to the usual semiconductor structure with a finite energy gap between the valence band and the conduction band. The presence in solid solutions of substitution atoms (Mn) with an uncompensated magnetic moment leads to a significant change in the band spectrum in a magnetic field, which depends on the state of the magnetic subsystem of the impurity ions. All this gives rise to a very specific behavior of the kinetic coefficients in charge-transport phenomena in the region of strong magnetic fields and of magnetooptical phenomena at low temperatures. Such "anomalous" properties as the splitting of the peaks of the Shubnikov-de Haas oscillations as the temperature is increased, the nonmonotonic temperature dependence of the amplitudes of these peaks, the sensitivity to temperature of the spectra of magneto-absorption and other phenomena can be successfully explained only by taking into account the magnetization of the band charge carriers by the molecular field of the localized electrons of the Mn ions. The magnetic properties of semimagnetic semiconductors have also turned out to be quite unique. Thus, in these crystals a concentration transition is observed from a paramagnetic phase into a spin-glass phase. The review presents the most important and significant results of the investigations of gapless and narrow-gap semimagnetic semiconductors obtained recently.