In this chapter, first we consider the diffeomorphism group Diff M of a general C∞ manifold M and introduce the concept of the diffeotopy group of M, which is by definition the group of isotopy classes of elements of Diff M. We also consider certain subgroups of Diff M consisting of those diffeomorphisms which preserve a given volume form or a symplectic form on M.
Then we specialize to the particular case where M is a closed oriented surface. In this case, the orientation preserving diffeotopy group is usually called the mapping class group. We also consider several groups related to the mapping class groups of surfaces such as the Torelli groups, the (outer) automorphism groups of free groups, the IA automorphism groups and the symplectomorphism groups of surfaces. We give some of the basic results concerning the algebraic structure of these groups. Then we would like to present a few problems for future research.