This review discusses the results of theoretical and experimental studies of biphonons (phonons bound to one another) and other effects of strong anharmonicity for the spectral region of overtones and compound tones of high-frequency optical vibrations of the crystal lattice. Longitudinal-transverse splitting of a biphonon and features of the structure of its spectrum in anisotropic crystals and Fermi resonance with bulk and surface polaritons are discussed, as well as the effect of this resonance on the dispersion of polaritons. Considerable attention is paid to analyzing the experimental data (infrared and Raman spectra) in which biphonons and cardinal rearrangement of the dispersion of a polariton in the region of a Fermi resonance have been found. The influence of a Fano antiresonance and singular points of the density of states are discussed in connection with Raman spectra of polaritons.