Abstract: I will consider the Dirichlet Laplacian $H_\theta$ in a twisted waveguide with rotational angle $\theta$ depending on the longitudinal variable. First, I will discuss the case where the derivative $\theta'$ can be written as $\beta - \epsilon$ with a constant $\beta > 0$, and a decaying function $\epsilon \geq 0$, and will discuss the asymptotic distribution of the discrete spectrum of $H_\theta$ near the bottom of its essential spectrum. Further, I will show that the wave operators for the operator pair $(H_\theta_1, H_\theta_2)$ exist and are complete, provided that the derivative $\theta'_1 - \theta'_2$ decays fast enough at infinity. Finally, I will assume that $\theta' = \beta - \epsilon$ with a real constant $\beta$, and a real function $\epsilon$ which decays fast enough at infinity. Using appropriate Mourre estimates, I will show that the singular continuous spectrum of $H_\theta$ \$ is empty. The talk will be based on joint works with Ph. Briet, H. Kovarik, and E. Soccorsi.