Stein's method and characteristic functions

We present a survey of various application of the method of the description of approximating distribution by means of differential equations for characteristic functions and, in particular, of applications of this description to estimates for the proximity of distributions. This idea was originally put forward by the author in 1976. Subsequently, this approach, which is called the Stein-Tikhomirov method by some authors (for instance, see papers by Eichelsbacher, Rednoss, Sunklodas, and Formanov), was significantly developed. Bibliography: 78 titles.