Actions of large finite groups on manifolds

In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group G on a manifold X, these results provide information on the restriction of the action to a subgroup of G of index bounded above by a number depending only on X. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology.