  We prove that if $F$ is a finitely generated abelian group of orientation
preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set
then there is a common fixed point for all elements of $F.$ We also show that
if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms
of $S^2$ then there is a common fixed point for all elements of a subgroup of
$F$ with index at most two.
