Mukai found the relation between polarized symplectic automorphism groups and certain subgroups of the Mathieu group. After the discovery of Mathieu moonshine, Huybrechts established the relation between autoequivalence groups of derived categories of K3 surfaces and certain subgroups of the Conway group. Although we know explicit examples of polarized K3 surfaces with maximal symplectic automorphism groups, it is difficult to find explicit examples of finite autoequivalences of derived categories of K3 surfaces not conjugate to automorphisms of K3 surfaces. In this talk, I would like to study how to construct finite autoequivalences of derived categories of K3 surfaces and discuss their difficulties. First, we recall automorphism groups of compact Riemann surfaces from point of view of algebraic geometry, topology and derived categories. Second, I would like to discuss autoequivalence groups of derived categories of K3 surfaces as an analogue of the case of compact Riemann surfaces.