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.