**Abstract: ** We consider Schr\"odinger operators $H=-\Delta+V$ in $L_2(\Omega)$ where the domain $\Omega\subset\R^{d+1}$ and the potential $V=V(x,y)$ are periodic with respect to the variable $x\in\R^d$. We assume that $\Omega$ is unbounded with respect to the variable $y\in\R$ and that $V$ decays with respect to this variable. $V$ may contain a singular term supported on the boundary. We develop a scattering theory for $H$ and present an approach to prove absence of singular continuous spectrum. Moreover, we show that certain repulsivity conditions on the potential and the boundary of $\Omega$ exclude the existence of surface states. In this case, the spectrum of $H$ is purely absolutely continuous and the scattering is complete. This is a joint work with R. Shterenberg.