Acoustic anisotropy is a consequence of anisotropy of the mechanical characteristics of a solid. In metals, it is associated with microstructural anisotropy of mechanical characteristics, internal mechanical stresses and strains, including residual stresses and plastic deformations. Sensors measuring acoustic anisotropy do not require complex preparations of a metal surface, therefore it is easy to measure which makes it possible for measurement results to be used to quantify stresses and strains in metals based on the magnitude of phase shifts of the shear wave velocities of the orthogonal polarization. Acoustic anisotropy is one of the manifestations of the phenomenon of changes in the elastic properties of an acoustic medium caused by mechanical stresses and deformation (acoustoelastic effect). This makes it possible to use the effect of acoustic anisotropy for the development of quantitative methods of acoustic tensometric measurements, as well as methods of non-destructive testing, which enables effective quality controls and diagnostics of the residual life of structures and machine parts. The article describes the history of the discovery and theoretical substantiation of the acoustoelastic effect and the quantitative relationship of acoustic anisotropy with stresses and deformations, starting with the pioneering works of the twentieth century. The way of forming the theory based on nonlinear mechanics of continuous media is shown. The third part of the article is concerned with an overview of the current state of research. An analysis is presented of experimental works on the measurement of acoustic anisotropy in low- and high-carbon steels, aluminum alloys, as well as in composites and other structural materials. Special attention is paid to a review of studies on the relationship between acoustic anisotropy and plastic deformations and the applicability limitations of the acoustic method. It also provides a list of the main applied results related to the measurement and use of acoustic anisotropy to control the blades of compressors and gas turbine engines, pipe steels, welded joints, etc. A review is given of the main publications on system analysis and generalization of theoretical and experimental scientific results obtained by domestic and foreign researchers in the field of studying the acoustic anisotropy of metallic structural materials under conditions of uniaxial and complex stress states, plastic deformation, thermomechanical loading and fatigue fracture is given.