  We apply an equivariant version of Perelman's Ricci flow with surgery to
study smooth actions by finite groups on closed 3-manifolds. Our main result is
that such actions on elliptic and hyperbolic 3-manifolds are conjugate to
isometric actions. Combining our results with results by Meeks and Scott [17],
it follows that such actions on geometric 3-manifolds (in the sense of
Thurston) are always geometric, i.e. there exist invariant locally homogeneous
Riemannian metrics. This answers a question posed by Thurston in [32].
