The fundamental results of recent research on the propagation of elastic microwaves (hypersonic waves) in crystals of various types are considered. The procedure for experiments with hypersonic waves is briefly described. The characteristics of the propagation of elastic waves in crystals (polarization, velocity, energy flux) and methods of their calculation are given. The principal mechanism of damping of hypersonic waves are considered, such as interactions with thermal phonons by the Akhiezer mechanism and by the Landau-Rumer mechanism, and interactions with free carriers and with defects. A separate chapter is devoted to problems of the propagation of hypersonic waves in magnetically-ordered crystals and their interactions with spin waves. The scattering of light by hypersonic waves is also considered, the singularities of this phenomenon are discussed, as is its use for the investigation of the characteristics of elastic-wave propagations. Some conclusions are drawn from the point of view of the prospects of using hypersonic methods in solid-state physics research.