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 of the group of smooth diffeomorphisms of an odd dimensional sphere admits no nontrivial homomorphisms to the group of diffeomorphisms of a ball of any dimension. This result generalizes theorems of Ghys and Herman. We also examine finitely generated subgroups of diffeomorphisms of spheres, and produce an example of a finitely generated torsion-free group with an action on the circle by smooth diffeomorphisms that does not extend to a C¹ action on the disc.