A theory to explain dark matter, dark energy, the evolution of the early Universe and other unsolved problems must correctly reproduce existing observational and experimental data, including the properties of gravitational waves. That is why we discuss the main properties of gravitational waves: the propagation speed, characteristic amplitudes, polarizations and other features in extended theories of gravity: scalar-tensor gravity, including the Brans-Dicke models, Horndeski theories, f(R)-gravity, theories with massive graviton, modified Newtonian dynamics, quantum theories of gravity, including loop quantum gravity, string theory IIB, Horava-Lifshitz theory, in higher-dimensional theories, including the general case of D-dimensional spacetime and Kaluza-Klein theory, in non-Riemannian geometry, including both the general case of metric affine gravity and various versions of teleparallel gravity: f(T) and f(Q), as well as the extended cases f(T,B) and f(T,T_G). With increasing experimental accuracy the discussed characteristics of gravitational waves will be discovered, helping in the selection of gravity theories.