There are many phenomena where algebraic geometry and group theory interplay. Many algebraic varieties have apparent or hidden symmetries such as ADE type of degenerate elliptic curves. In 80's I classified finite groups acting symplectically on a K3 surface, in terms of the Mathieu group M24, a sporadic simple finite groups. I will survey an analogous result which I and H. Ohashi recently obtained for Enriques surfaces, another class of algebraic surfaces of Kodaira dimension 0. As a second sample of hidden symmetries, I will discuss a certain connection between Fano threefolds and four Dynkin diagrams of exceptional types.