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 \(W^{n+1,p}(U)\).