studies of high contrast optical systems[^4] used to model photonic crystals — see, e.g. [@FK; @KK] — which in a suitable approximation yield an analogue of the spectral problem for the operator (\[formal\]); the two differ only by the physical interpretation, the roles of the coupling and spectral parameters being switched. A derivation of leaky-graph models in this context was given in the paper [@FK2], see also the review [@Ku] and recall that the corresponding operators can be cast also in a pseudo-differential form [@FK2; @PP].

The material we are going to review is relatively extensive. We will take care, of course, to explain properly all the notions and the results. On the other hand, proofs will be mostly sketched. However, we will always give references to original papers where the particular complete argument can be found. Let us finally remark that the subject reviewed here cannot be regarded as closed, on the contrary, there are many open questions. We devote to them the closing section, and the author of this survey can only hope that his reader will take this problem list as a challenge and a program which will keep him or her busy for some time.

Leaky 