  We discuss asymptotics for large random planar maps under the assumption that
the distribution of the degree of a typical face is in the domain of attraction
of a stable distribution with index $\alpha\in(1,2)$. When the number $n$ of
vertices of the map tends to infinity, the asymptotic behavior of distances
from a distinguished vertex is described by a random process called the
continuous distance process, which can be constructed from a centered stable
process with no negative jumps and index $\alpha$. In particular, the profile
of distances in the map, rescaled by the factor $n^{-1/2\alpha}$, converges to
a random measure defined in terms of the distance process. With the same
rescaling of distances, the vertex set viewed as a metric space converges in
distribution as $n\to\infty$, at least along suitable subsequences, toward a
limiting random compact metric space whose Hausdorff dimension is equal to
$2\alpha$.
