  This text is a survey (Bourbaki seminar) on the paper "Liouville quantum
gravity and KPZ" By B.Duplantier and S.Sheffield.
  The study of statistical physics models in two dimensions (d=2) at their
critical point is in general a significantly hard problem (not to mention the
d=3 case). In the eighties, three physicists, Knizhnik, Polyakov et
Zamolodchikov (KPZ) came up in \cite{\KPZ} with a novel and far-reaching
approach in order to understand the critical behavior of these models. Among
these, one finds for example random walks, percolation as well as the Ising
model. The main underlying idea of their approach is to study these models
along a two-step procedure as follows: a/ First of all, instead of considering
the model on some regular lattice of the plane (such as $\Z^2$ for example),
one defines it instead on a well-chosen "random planar lattice". Doing so
corresponds to studying the model in its {\it quantum gravity} form. In the
case of percolation, the appropriate choice of random lattice matches with the
so-called planar maps. b/ Then it remains to get back to the actual {\it
Euclidean} setup. This is done thanks to the celebrated {\bf KPZ formula} which
gives a very precise correspondence between the geometric properties of models
in their quantum gravity formulation and their analogs in the Euclidean case.
  The nature and the origin of such a powerful correspondence remained rather
mysterious for a long time. In fact, the KPZ formula is still not rigorously
established and remains a conjectural correspondence. The purpose of this
survey is to explain how the recent work of Duplantier and Sheffield enables to
explain some of the mystery hidden behind this KPZ formula. To summarize their
contribution in one sentence, their work implies a beautiful interpretation of
the KPZ correpondence through a uniformization of the random lattice, seen as a
Riemann surface.
