The mapping class group is an important algebraic invariant of a topological space and an important object in low-dimensional geometry and topology, especially in the geometry and topology of 3-manifolds. In this lecture, we will first introduce and review some material about mapping class groups. Then, we will review theorems that explain the structure of these groups and their relations to the Teichmüller spaces. After that, we will discuss some applications of this material in the theory of Riemann surfaces and number theory, especially Grothendieck-Teichmüller theory.