Аннотация:A survey of the development of the traveling wave method for one-dimensional mediais presented. The main results and changes in the statement of the problem of representingsolutions of linear systems of partial differential equations in terms of “traveling waves” (moreprecisely, in terms of a system of wave transport equations) are presented. It is shown that asthe study of systems becomes more complicated, the problem of representing the solution bythe traveling wave method turns out to be applicable not only for hyperbolic systems but alsofor systems containing (even implicitly) both parabolic and elliptic components and therebyapproaches the general problem of decomposition of an arbitrary system of linear equations intoa system of first-order equations with a main part of the canonical type and with a subordinatelinear part.