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

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