Finite groups of birational selfmaps

Groups of birational selfmaps of algebraic varieties may be large and difficult to understand. On the other hand, the structure of their finite subgroups is often much more accessible. One example is given by K3 surfaces, whose automorphism group may be infinite but always contains just a finite number of finite subgroups up to isomorphism. I will survey the results on boundedness of finite groups acting by automorphisms and birational automorphisms of algebraic varieties, and discuss other examples with similar properties.