2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/145304The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-Δ-αδ(x-Γ)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^ν,\: ν=2,3$. We will explain how such singular Schrödinger operators can be properly defined for different codimensions of $Γ$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $Γ$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.42 pages, to appear in the proceedings of the Isaac Newton Institute programme "Analysis on Graphs and Applications"; the final version with minor improvementsMathematical PhysicsMesoscale and Nanoscale PhysicsAnalysis of PDEsSpectral TheoryQuantum Physics81Q05 (Primary); 35J10, 35P15, 58J05 (Secondary)text