  We discuss scaling limits of large bipartite quadrangulations of positive
genus. For a given $g$, we consider, for every $n \ge 1$, a random
quadrangulation $\q_n$ uniformly distributed over the set of all rooted
bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric
space by endowing its set of vertices with the graph distance. We show that, as
$n$ tends to infinity, this metric space, with distances rescaled by the factor
$n^{-1/4}$, converges in distribution, at least along some subsequence, toward
a limiting random metric space. This convergence holds in the sense of the
Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of
the choice of the subsequence, the Hausdorff dimension of the limiting space is
almost surely equal to 4. Our main tool is a bijection introduced by Chapuy,
Marcus, and Schaeffer between the quadrangulations we consider and objects they
call well-labeled $g$-trees. An important part of our study consists in
determining the scaling limits of the latter.
