Moduli of families of curves and quasiconformal mappings (Q1889946)

This paper is an extensive survey based on the author's 1997 Dr.Sci. dissertation in Novosibirsk State University. It reviews several applications of the well known method of moduli of families of curves on a Riemann surface due to many authors, with an extensive bibliography. The main sections of of this survey are: 1. Moduli of free families of homotopy classes of curves and their properties; 2. Moduli in extremal problems for quasiconformal mappings of Riemann surfaces; 3. Applications to Teichmüller theory and metrics on Teichmüller space.