"Methods of the Classical Theory of Elastodynamics" deals not only with classical methods as developed in the past decades, but presents also very recent approaches. Applications and solutions to specific problems serve to illustrate the theoretical presentation. Keywords: Smirnov-Sobolev method with further developments; integral transforms; Wiener-Hopf technique; mixed boundary-value problems; time-dependent boundaries; solutions for unisotropic media (Willis method); 3-d dynamical problems for mixed boundary conditions.