Studies on cubically nonlinear elastic waves are reviewed: the state of affairs in research on cubically and quadratically nonlinear waves in physics is briefly reported; a general view on nonlinear elastic waves is formulated; and mainly hyperelastic materials are considered. A description is given to well-known elastic potentials and various modifications of the Murnaghan potential such as the classical modification (in five forms), Guz's modification, Mindlin–Eringen modification, modification for a Cosserat pseudocontinuum, modification for Le Roux gradient theory, and two modifications for the theory of elastic mixtures. A procedure of transition from the potential to wave equations is described; and the corresponding wave equations for plane polarized waves are written for all the modifications mentioned above. Three basic methods for solving wave equations are considered and commented on: the method of successive approximations, the method of slowly varying amplitudes, and a wavelet-based method. The last method is discussed in more detail and exemplified