Nonlinear optical phenomena are widely used for the study of semiconductor materials. The paper presents an overview of experimental and theoretical studies of excitons by the method of optical second and third harmonics generation in various bulk semiconductors (GaAs, CdTe, ZnSe, ZnO, Cu$_2$O, (Cd,Mn)Te, EuTe, EuSe), and low-dimensional heterostructures ZnSe/BeTe. Particular attention is paid to the role of external electric and magnetic fields that modify the exciton states and induce new mechanisms of optical harmonics generation. Microscopic mechanisms of harmonics generation based on the Stark effect, the spin and orbital Zeeman effects, and on the magneto-Stark effect specific for excitons moving in an external magnetic field are considered. This approach makes it possible to study the properties of excitons and to obtain new information on their energy and spin structure that is not available when the excitons are investigated by linear optical spectroscopy. As a result of these studies, a large amount of information was obtained, which allows us to conclude on the establishing of a new field of research - exciton spectroscopy by the method of optical harmonics generation.
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