Let \phi be a quasiconformal mapping, and let T_\phi be the composition operator which maps f to f\circ\phi . Since \phi may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of T_\phi on L^p and W^{1,p} for 1 < p < \infty . This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in H^{s,p} are sent to H^{s,q} whenever 0 < s < 1 for appropriate values of q . The techniques used lead to sharp results and they can be applied to Besov spaces as well.