The aim of this review article is to collect together separated results of research in the application of fractional derivatives and other fractional operators to problems connected with vibrations and waves in solids having hereditarily elastic properties, to make critical evaluations, and thereby to help mechanical engineers who use fractional derivative models of solids in their work. Since the fractional derivatives used in the simplest viscoelastic models (Kelvin-Voigt, Maxwell, and standard linear solid) are equivalent to the weakly singular kernels of the hereditary theory of elasticity, then the papers wherein the hereditary operators with weakly singular kernels are harnessed in dynamic problems are also included in the review. Merits and demerits of the simplest fractional calculus viscoelastic models, which manifest themselves during application of such models in the problems of forced and damped vibrations of linear and nonlinear hereditarily elastic bodies, propagation of stationary and transient waves in such bodies, as well as in other dynamic problems, are demonstrated with numerous examples. As this takes place, a comparison between the results obtained and the results found for the similar problems using viscoelastic models with integer derivatives is carried out. The methods of Laplace, Fourier and other integral transforms, the approximate methods based on the perturbation technique, as well as numerical methods are used as the methods of solution of the enumerated problems. This review article includes 174 references.