Inverse problems of the plane waves scattering of inclined incidence of the SH type by an inhomogeneous half-space (in particular, by a transition layer) or by an inhomogeneous layer with a free boundary are considered. The characteristics of an inhomogeneous elastic medium, i.e., the wave propagation velocity *υ(z)* and the density *ρ(z)*, are functions of depth *z* and should be determined in the inverse problems. The following are considered: the inverse problems with data for a fixed angle of incidence *θ0* of a plane wave (the angle between the vector of the normal to the plane wave front and *z*-axis) when the shape *φ0(ξ,θ0)* of an incident wave is known; the inverse problems with data for a family of angles *θ0* both for known and unknown shapes of the incident wave. The functions *φ0(ξ,θ0)* and *φ1(ξ,θ0)* (the shapes of incident and reflected waves), the functions *φ0(ξ,θ0)* and *u(H,ξ,θ0)* (the shapes of the incident wave and the free boundary oscillation field), only the function *u(H,ξ,θ0)* (in the inverse problem with the unknown function *φ0(ξ,θ0)*, or other data are given as data corresponding to any fixed value of *θ0*. Possible application areas of the inverse problems under consideration are specified.

This paper is essentially a review. A new result presented here has been obtained for inverse problems with the data {*φ0(ξ,θ0)*, *φ1(ξ,θ0)*} or {*φ(ξ,θ0)*, *u(H,ξ,θ0)*} for a set of angles *θ0*. The results, received earlier, (the uniqueness theorem, the solution method) are extended to the case when the limiting point *Ō0* of this set is zero. Also, a new explicit formula, which makes it possible to find the functions *υ(z)* and *ρ(z)* using the data of these inverse problems for *Ō0* = 0, has been obtained.