A fat graph is, generally speaking, a family of spaces depending on a parameter $\varepsilon$ which converge metrically to a metric graph as $\varepsilon\to 0$. The problem of studying the behavior of the spectrum of the Laplacian or other geometric operators, under this limit, arises in various contexts and can be approached by a variety of techniques. In this survey talk I will explain some of these techniques and mention some recent results.
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