Dessins d’enfants and moduli spaces of curves

A very brief overview of the dessins d'enfants theory will be given. Two kinds of its relations with the moduli spaces of curves will be outlined. (1) The Mumford-Penner-Kontsevich-... construction of cell decomposition of (decorated) moduli spaces and its arithmetic version (Mulase-Penkava). (2) The critical filtrations of Hurwitz and moduli spaces, introduced by the speaker, will be provided, some examples given and open questions formulated.