"Random maps and their scaling limits" Gregory Miermont, Université Paris Sud and University of British Columbia In this talk, I will present some recent progress on the convergence of large random quadrangulations - i.e. a large uniform gluing of squares forming a topological sphere - towards the so-called Brownian map, which is a universal model for a continuum random surface. Proving this convergence, which holds in the Gromov-Hausdorff topology, requires a precise study of geodesics in large quadrangulations and in the limiting space, and in particular, of the locus where geodesics tend to separate. If time allows, I will also formulate some conjectures concerning O(n) loop models on random quadrangulation, and their relation with the so-called stable maps, which are scaling limits for random maps with large faces.