Moduli of Families of Curves for Conformal and Quasiconformal Mappings Introduction Moduli of Families of Curves and Extremal Partitions Moduli in Extremal Problems for Conformal Mapping Moduli in Extremal Problems for Quasiconformal Mapping. Moduli on Teichmüller Spaces References List of simbols Index. The monograph is concerned with the modulus of families of curves on Riemann surfaces and its applications to extremal problems for conformal quasiconformal mappings and the extension of the modulus onto Teichmüller spaces. The main part of the monograph deals with extremal problems for compact classes of univalent conformal and quasiconformal mappings. Many of them are grouped around two point distortion theorems. Montels functions and functions with fixed angular derivatives are also considered. The last portion of problems is directed to the extension of the modulus varying the complex structure of the underlying Riemann surface that sheds some new light on the metric problems of Teichmüller spaces.