Abstract: In this lecture we define self-adjoint Schrödinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface explicitly via boundary conditions. Some qualitative spectral properties of these operators are investigated and a variant of Krein's formula is shown. Furthermore, Schatten-von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with $\delta$ and $\delta'$-potentials, and the Schrödinger operator without a singular interaction are discussed. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the spectra of the singularly perturbed and unperturbed Schrödinger operators. The talk is based on joint work with Matthias Langer (University of Strathclyde, Glasgow) and Vladimir Lotoreichik (Graz University of Technology).