In this thesis, we study the differences between smooth and topological manifolds in the equivariant setting. The central topic will be on smooth structures. Kirby-Siebenmann showed (using work of Kervaire-Milnor) that every closed high dimensional manifold has only finitely many smooth structures.In contrast, if G is a nontrivial finite group, we construct G-manifolds with infinitely many equivariant smooth structures. Our examples even include nonpositively curved manifolds on which G acts by isometries.