  The standard actions of finite groups on spheres S^d are linear actions, i.e.
by finite subgroups of the orthogonal group O(d+1). We prove that, in each
dimension d>5, there is a finite group G which admits a faithful, topological
action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The
situation remains open for smooth actions.
