Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other results in the literature that allow for sampling other classes of planar maps, but I haven't seen any results for the entire class of planar maps.
Something you may potentially be interested in : http://www.lix.polytechnique.fr/~fusy/Articles/FusyAofa.pdf
Planar quadrangulation with $n$ faces are in bijection with planar maps with $n$ edges. The quadrangulation is bi-partite, so colour its vertices white/black. inside each face, add a diagonal edge between the two black vertices. Then delete all white vertices and incident edges. The result is a map with $n$ edges, and it is not hard to see that this is a bijection.
In fact, this is one of the reasons quadrangulations are a particularly interesting class of planar maps.