The curve graph of a surface of finite type is a graph that encodes the combinatorics of isotopy classes of simple closed curves. It is a fundamental tool for the study of the geometric group theory of the mapping class group. In 1987 N.K. Ivanov proved that the automorphism group of the curve graph of a finite surface is the extended mapping class groups. In the following decades, many people proved analogue results for many "similar" graphs, such as the pants graph, the arc graph, etc. In response to the many results, N.V. Ivanov formulated a metaconjecture, which asserts that any "natural and sufficiently rich" object associated to a surface has automorphism group isomorphic to the extended mapping class group. In this talk, I will present a joint work with Thomas Koberda (Virginia) and Javier de la Nuez Gonzalez (KIAS) where we provide a model theoretical framework for Ivanov's metaconjecture and we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different "similar" geometric complexes. In particular, we will prove that the curve graph of a surface of finite type is w-stable.