the situation where the attraction is position independent, $\alpha(x)=\alpha>0$. Before proceeding to a definition of such singular Schrödinger operators and discussion of their properties, let us make a few remarks.

First of all, it is clear that there is no ambiguity related to the graph vertices once $\Gamma$ and $\alpha$ are given. It is equally obvious that the confinement in this model takes place at negative energies only. The particle now “lives” in the whole space and can be found even at large distances from $\Gamma$, although with a small probability, because the complement $\mathbb{R}^\nu \setminus\Gamma$ is the classically forbidden region. The presence of the tunneling is the reason why we dub such systems as *leaky quantum graphs*.

Schrödinger operators with interactions supported by curves and other manifolds of a lower dimension were studied already in the early nineties [@BT], and even earlier in examples with a particular symmetry [@AGS; @Sha]. A more systematic investigation motivated by the above considerations was undertaken in a last few years; it is the aim of this review to describe its results.

One should stress, however, that such mathematical structures can be studied also from other points of view. A prominent example comes from 