Abstract: In this talk we illustrate how abstract methods from extension theory of symmetric operators can be applied in the spectral analysis of elliptic partial differential operators. In this context we discuss results on Schrödinger operators with $\delta$-interactions on hypersurfaces, spectral estimates and asymptotics of the difference of selfadjoint realizations of elliptic PDEs, Dirichlet-to-Neumann maps and their relation to the spectrum, as well as selfadjoint realizations and maximal trace maps for the Laplacian on bounded Lipschitz domains.