  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.
