Singular Schr\"odinger operators and Robin billiards. Spectral properties and asymptotic expansions

This paper summarizes the contents of a plenary talk at the Pan African Congress of Mathematics held in Rabat in July 2017. We provide a survey of recent results on spectral properties of Schr\"odinger operators with singular interactions supported by manifolds of codimension one and of Robin billiards with the focus on the geometrically induced discrete spectrum and its asymptotic expansions in term of the model parameters.