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The development of linear elastodynamics in pure stress-based formulation began over half-a-century ago as an alternative to the classical displacement-based treatment that came into existence two centuries ago in the school of mathematical physics in France. While the latter approach – fundamentally based on the Navier displacement equation of motion – remains the conventional setting for analysis of wave propagation in elastic bodies, the stress-based formulation and the advantages it offers in elastodynamics and its various extensions remain much less known. Since the key mathematical results of that formulation, as well as a series of applications, originated with J. Ignaczak in 1959 and 1963, the key relation is named the Ignaczak equation of elastodynamics. This review article presents the main ideas and results in the stress-based formulation from a common perspective, including (i) a history of early attempts to find a pure stress language of elastodynamics, (ii) a proposal to use such a language in solving the natural traction initial-boundary value problems of the theory, and (iii) various applications of the stress language to elastic wave propagation problems. Finally, various extensions of the Ignaczak equation of elastodynamics focused on dynamics of solids with interacting fields of different nature (classical or micropolar thermoelastic, fluid-saturated porous, piezoelectro-elastic) as well as nonlinear problems are reviewed. [ABSTRACT FROM AUTHOR]