The action of conformal transformations on a Riemannian manifold.
Answering a question of Smale, we prove that the space of C 1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.
For actions as in the title we associate a collection of rotation numbers. If one of them is sufficiently irrational then the action is conjugate (as an action) to either a linear action on a torus or to an action on a principal bundle over with orbits.