  In this paper we discuss the relationship between groups of diffeomorphisms
of spheres and balls. We survey results of a topological nature and then
address the relationship as abstract (discrete) groups. We prove that the
identity component Diff_0(S^{2n-1}) of the group of smooth diffeomorphisms of
S^{2n+1} admits no nontrivial homomorphisms to the group of C^1 diffeomorphisms
of the ball B^m for any n and m. This result generalizes theorems of Ghys and
Herman.
  We also examine finitely generated subgroups of Diff_0(S^n) and produce an
example of a finitely generated torsion free group Gamma with an action on the
circle by smooth diffeomorphisms that does not extend to a C^1 action of Gamma
on the disc.
