A review of papers dealing with the investigation of wave phenomena occurring in weakly anisotropic elastic media is available in the present article. Several weakly anisotropic models of a continuous medium are presented and discussed which can be used to advantage both for solving the problems of wave dynamics and for carrying out comparison analysis of experimental data on measurements of elastic moduli of slightly anisotropic material by quasilongitudinal and quasitransverse waves with the results of the theoretical investigations. Based on these models, the behavior of harmonic and nonstationary bulk and surface waves and shock waves propagating in linear and nonlinear elastic isotropic and anisotropic media are examined. The problems are solved by the perturbation technique. The value characterizing the deviation of anisotropic elastic coefficients from isotropic moduli is chosen as a small parameter. Straight expansions of the desired values with respect to the small parameter are obtained, and the domains of nonuniformity of such expansions are indicated. For some of the problems under consideration, it has been possible to regularize the straight expansions of the values to be found by virtue of the especially selected manipulations of dilatation or translation of independent coordinates and to construct the uniformly valid solutions in the whole domain of wave motion existence. This review article contains 280 references.