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 Formula: see text on a manifold Formula: see text, these results provide information on the restriction of the action to a subgroup of Formula: see text of index bounded above by a number depending only on Formula: see text. 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.