We consider a twisted quantum waveguide i.e. a domain of the form ��_�� : = r_����\times R, where ��\subset R^2 is a bounded domain, and r_��= r_��(x_3) is a rotation by the angle ��(x_3) depending on the longitudinal variable x_3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H_��, self-adjoint in L^2 (��_��), and consider related scattering problems. First, we show that if the derivative of the difference ��_1 - ��_2 decays fast enough as |x_3| goes to infinity, then the wave operators for the operator pair (H_{��_1}, H_{��_2}) exist and are complete. Further, we concentrate on appropriate perturbations of constant twisting, i.e. ��' = ��- ��, with constant ��\in R, and ��which decays fast enough at infinity together with its first derivative. In this case the unperturbed operator corresponding to ��is an analytically fibered Hamiltonian with purely absolutely continuous spectrum. Obtaining Mourre estimates with a suitable conjugate operator, we prove, in particular, that the singular continuous spectrum of H_��, is empty.

28 pages; typos corrected; to appear in Journal of Functional Analysis