Mapping Class group of a closed surface $\Sigma$ is defined as the path component of homeomorphism group of $\Sigma$ containing the identity homeomorphism. The theory of mapping class groups of surfaces is very rich and has wide variety of applications in understanding the geometry of surfaces. We would go through the classical theorems of mapping class groups of finite type surfaces like Dehn-Lickorish theorem, Dehn-Nielsen-Baer theorem and then try to understand their infinite-type counterparts (if any). We might also prove the fact that the action of Mapping class group on Teichmuller space is properly discontinuous.