Some results and conjectures on finite groups acting on homology spheres

This is a note based on a talk given in the Workshop on geometry and topology of $3$-manifolds, Novosibirsk, 22–26 August 2005. We consider the class of finite groups, which admit arbitrary, i.e. not necessarily free actions on integer and $\bmod2$ homology spheres, with an emphasis on the $3$- and $4$-dimensional cases. We recall some classical results and present some recent progress as well as new results, open problems and the emerging conjectural picture of the situation.