ABSTRACT
This chapter explains quasiconformal mappings between Riemann surfaces of finite analytic type. It describes the theorems involved in the quasiconformal mappings, quadrilaterals mappings, and quasiconformal mappings. Symmetric quasicircles was introduced by Becker and Pommerenke as asymptotically conformai curves. They gave a number of analytical characterizations for symmetric quasicircles in terms of conformai mappings. Later Gardiner and Sullivan introduced the corresponding concept of symmetric homeomorphisms of the unit circle (or real line) from the topological point of view. It is well known that quasisymmetric homeomorphisms are boundary values of quasiconformal mappings and they can be characterized by the so called M-condition.
