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 and Zamolodchikov (KPZ) came up 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:
- First of all, instead of considering the model on some regular lattice of the plane (such as ${\mathbb Z}^2$ for example), one defines it instead on a well-chosen "random planar lattice''. Doing so corresponds to studying the model in its quantum gravity form. In the case of percolation, the appropriate choice of random lattice matches with the so-called planar maps which are currently the subject of an intense activity (for example the recent works by Le Gall and Miermont).

- Then it remains to get back to the actual Euclidean setup. This is done thanks to the celebrated 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 correspondence through a uniformization of the random lattice, seen as a Riemann surface.