Random planar maps (which correspond to planar graphs embedded in the plane) are a very natural model of random discrete surface, which have been widely studied in these last 30 years (in particular by several members of the Combi team !). In my talk, I will present some results of convergence — in the local limit sense, as introduced by Benjamini and Schramm – for those models. I will try to give an overview of this field: I will first consider models of maps which are sampled from a uniform distribution, for which many results are available. I will then move to random planar maps sampled with a weight which comes from a statistical physics model, for which many problems are still open and yield fascinating research perspectives.