Let [katex] \phi [/katex] be a quasiconformal mapping, and let [katex] T_\phi [/katex] be the composition operator which maps [katex] f [/katex] to [katex] f\circ\phi [/katex]. Since [katex] \phi [/katex] may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of [katex] T_\phi[/katex] on [katex]L^p [/katex] and [katex]W^{1,p} [/katex] for [katex]1 < p < \infty [/katex]. 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 [katex]H^{s,p} [/katex] are sent to [katex]H^{s,q} [/katex] whenever [katex]0 < s < 1 [/katex] for appropriate values of [katex]q [/katex]. The techniques used lead to sharp results and they can be applied to Besov spaces as well.