Abstract: In this talk we discuss a recent result on the representation of the scattering matrix in terms of an abstract Weyl function. The general result can be applied to scattering problems for Schrödinger operators with $\delta$-type interactions on hypersurfaces, and scattering problems involving Neumann and Robin realizations of Schrödinger operators on unbounded domains. In both applications we obtain formulas for the corresponding scattering matrices in terms of Dirichlet-to-Neumann maps. This talk is based on joint work with Mark Malamud and Hagen Neidhardt.