  We consider large random planar maps and study the first-passage percolation
distance obtained by assigning independent identically distributed lengths to
the edges. We consider the cases of quadrangulations and of general planar
maps. In both cases, the first-passage percolation distance is shown to behave
in large scales like a constant times the usual graph distance. We apply our
method to the metric properties of the classical Tutte bijection between
quadrangulations with $n$ faces and general planar maps with $n$ edges. We
prove that the respective graph distances on the quadrangulation and on the
associated general planar map are in large scales equivalent when $n \to
\infty$.
