The main results of this dissertation concern the structure of the group of diffeomorphisms of a smooth manifold. Filipckiewicz's theorem states that two manifolds M and N have isomorphic diffeomorphism groups (isomorphic as abstract groups) if and only if the manifolds M and N are diffeomorphic. We give a new proof of Filipckiewicz's result and establish a generalization of it by showing that any non-trivial abstract homomorphism between groups of diffeomorphisms is in some sense continuous. The proof of our results are related to the concept of distortion in geometric group theory, concept which has proven to be useful for the study and classification of certain group actions on manifolds. We also give new dynamical restrictions that certain type of distorted elements in diffeomorphism groups known as ``arbitrarily distorted" must have.
