The focus of this review article is on analytical procedures and physical effects which are characteristic of the theory of nonlinear and simple waves in materials. Waves are supposed to propagate in composite materials, which are modeled as solid two-phase mixtures. It is shown how procedures of wave interaction investigations in nonlinear acoustics, optics and radiophysics are applied to nonlinear mechanics of materials with a microstructure. Main effects of the interactions of waves in composite materials: new harmonics generation, self-generation, evolution and distortion, synchronization, breakdown instability, etc are commented upon. This article is proposed not only for specialists in wave theories; therefore it contains some facts which are obvious for researchers working in the field of waves. Many portions of this review are described in more detail in a book (Rushchitsky and Tsurpal (1998), 377 pages). This review article contains 286 references.