**Abstract:** In this talk I am going to discuss Schr\"odinger operators with an attractive singular `potential' supported by a manifold of a lower dimensionality. Formally one can write them as $-\Delta-\alpha \delta(x-\Gamma)$ with $\alpha>0$, where $\Gamma$ is a curve in $\mathbb{R}^d,\: d=2,3$, or a surface in $\mathbb{R}^3$. After illustrating that the geometry of $\Gamma$ can give rise to an effective interaction, we concentrate on the strong-coupling asymptotic behaviour. We show that after the natural energy renormalization, the leading term is given by an operator on $\Gamma$ with a geometrically induced potential. We discuss the mentioned dimension and codimension situations including periodic manifolds and magnetic Schr\"odinger operators. In the simplest case of a planar arc we also prove a conjecture about the asymptotic behaviour for manifolds with a boundary.