In mathematical physics, the inverse scattering problem determines the characteristics of an object based on how it scatters radiation or particles. The approach is the inverse of the "direct scattering problem," which investigates how particles are distributed based on characteristics of the scattering object. This monograph by two prominent Soviet authorities in mathematical physics was a major contribution to the literature in this field. The two-part treatment begins by examining the boundary-value problem without singularities. Topics include particular solutions of the system without singularities, the spectrum and scattering matrix for the boundary-value problem without singularities, Parseval's equality, and the inverse problem. The second part addresses the boundary-value problem with singularities, exploring special transformation operators, spectral analysis, and reconstruction of the problem from scattering data. Helpful Appendixes supplement the text. For advanced undergraduates and graduate students in physics and related fields.