  We introduce bijections between families of rooted maps with unfixed genus
and families of so-called blossoming trees endowed with an arbitrary forward
matching of their leaves. We first focus on Eulerian maps with controlled
vertex degrees. The mapping from blossoming trees to maps is a generalization
to unfixed genus of Schaeffer's closing construction for planar Eulerian maps.
The inverse mapping relies on the existence of canonical orientations which
allow to equip the maps with canonical spanning trees, as proved by Bernardi.
Our bijection gives in particular (here in the Eulerian case) a combinatorial
explanation to the striking similarity between the (infinite) recursive system
of equations which determines the partition function of maps with unfixed genus
(as obtained via matrix models and orthogonal polynomials) and that determining
the partition function of planar maps. All the functions in the recursive
system get a combinatorial interpretation as generating functions for maps
endowed with particular multiple markings of their edges. This allows us in
particular to give a combinatorial proof of some differential identities
satisfied by these functions. We also consider face-colored Eulerian maps with
unfixed genus and derive some striking identities between their generating
functions and those of properly weighted marked maps. The same methodology is
then applied to deal with $m$-regular bipartite maps with unfixed genus,
leading to similar results. The case of cubic maps is also briefly discussed.
