This is a survey on old and new results as well as
an introduction to various related basic notions and concepts, based on two talks given at the International Workshop on Geometry and Analysis in Kemerovo (Sobolev Institute of Mathematics, Kemerovo State University) and at the University of Krasnojarsk in June 2011. We discuss finite groups acting on low-dimensional spheres and homology spheres, comparing with the finite subgroups of the corresponding orthogonal groups, and also finite simple groups acting on spheres and homology spheres of arbitrary dimension.
