A brief survey of soliton (solitary-wave) solutions to various nonlinear PDEs that arise in mechanics of continuum elastic media is given. The solutions are actually described by ODEs, obtained from the corresponding PDEs for a function \(u(x,t)\) by the substitution \(u(x,t) = u(x - Vt)\), where \(V\) is the soliton’s velocity. The soliton corresponds to a homoclinic solution to this ODE. Only simple (single-humped) solitons are possible if the nonlinear ODE is of the second order (which can be obtained, e.g., from the sine-Gordon or nonlinear Schrödinger partial differential equations). If the effective ODE is of the fourth order, which may be obtained from more sophisticated PDEs, infinitely many higher-order solitons with an arbitrary integer number of the humps exist.