  The enumeration of maps and the study of uniform random maps have been
classical topics of combinatorics and statistical physics ever since the
seminal work of Tutte in the sixties. Following the bijective approach
initiated by Cori and Vauquelin in the eighties, we describe a bijection
between rooted maps, or rooted bipartite quadrangulations, on a surface of
genus g and some simpler objects that generalize plane trees. Thanks to a
rerooting argument, our bijection allows to compute the generating series of
rooted maps on a surface of genus g with respect to the number of edges, and to
recover the asymptotic numbers of such maps.
  Our construction allows to keep track in a bipartite quadrangulation of the
distances of all vertices to a random basepoint. This is an analog for higher
genus surfaces of the basic result on which were built the recent advances in
the comprehension of the intrinsec geometry of large random planar maps,
hopefully opening the way to the study of a model of continuum random surfaces
of genus g.
