The theory of planar map enumeration was started by Tutte and iniciated the theory of map enumeration when trying to solve the 4-colour theorem by enumerative arguments. Nowadays a wide diversity of maps have been enumerated (by enumerated means to either get the exact formula, or generating functions), specially when the parameter is the number of edges.
The techniques nowadays are diverse: algebraic techniques (so-called catalytic variables), bijective methods and matrix integral methods.
My question is the following, and I restric myself to 1) families of maps without extra structure/statistical model (no Ising/Potts model, etc) and 2) maps on the sphere.
a) Which maps do we know to count? This is maybe a too general question, but for instance with respect to face degree (or, by duality, vertex degree), connectivity, etc.
b) Is there an updated survey stating what is known (and maybe more interesting, what is NOT known) concerning map enumeration?
