The problem of wave diffraction by wedge-shaped structures is important in many applications, including electromagnetics, acoustics and geophysics. Numerous attempts to solve these problems have been made during the last 70 years using both analytical and numerical methods, but this has proved to be very difficult; only a few of the simplest problems of this type have been successfully solved. The difficulties are caused by the nonseparability of variables and the singular behaviour of solutions near the vertex. This Research Note presents a systematic approach allowing the consideration of arbitrary problems of this type and the development of stable numerical algorithms for their solution.