In this dissertation some new results on the boundedness of Calderón-Zygmund operators on Sobolev spaces on domains in R^d. First a T(P)-theorem is obtained which is valid for W^{n,p} (U), where U is a bounded uniform domain of R^d, n is a given natural number and p>d. Essentially, the result obtained states that a convolution Calderón-Zygmund operator is bounded on this function space if and only if T(P) belongs to W^{n,p} (U) for every polynomial P of degree smaller than n restricted to the domain. For indices p less or equal than d, a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of n=1 and p \leq d, this Carleson condition is shown to be necessary in fact. The case where n is not integer and 0 < n < 1 is also studied, and analogous results to the former are obtained for a larger family of function spaces, the so-called Triebel-Lizorkin spaces. The thesis contains some optimal conditions to establish when the Beurling transform of a polynomial restricted to a domain is contained in a Sobolev space W^{n,p}(U), where U is a bounded planar lipschitz domain, in terms of the Besov regularity of the boundary of U. This result, in combination with the one mentioned above, provides a condition to determine whether the Beurling transform is bounded on W^{n,p}(U) or not for p>2, which is optimal in case n=1.

Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity W^{n,p}(U) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain U with boundary parameterizations in a certain Besov space and p>2, implies that the mapping itself is in W^{n+1,p}(U).