The results are presented of the ten-year development of the concepts of anharmonicity of the normal modes that describe the vibrations of a continuous medium having two or more mutually coupled subsystems. The results are discussed of experimental and theoretical studies of the anharmonicity of the magnetoelastic normal waves in antiferromagnetics having weak anisotropy and a large exchange field, such as $\alpha$ -Fe$_2$O$_3$, FeBO$_3$, etc. In these crystals the ``effective'' anharmonicity of the quasiacoustic branch of vibrations that arises from the magnetostrictive coupling of the elastic and spin subsystems is especially large--larger by a factor of 10$^3$ to 10$^4$ than that typical of a pure elastic medium. The corresponding effective moduli are controlled easily, and over a broad range, by a small magnetic field (0.1--2 kOe). The anomalously large (``giant'') magnitude of the acoustic nonlinearity has enabled observing many ultrasound analogs of the effects of nonlinear optics (frequency doubling of sound, stimulated combination scattering of sound by sound, etc.). Realization in the near future of other analogs--self-focusing of a sound wave, excitation of magnetoelastic solitons, etc., is expected.