Abstract: I will survey some recent progress on how to define natural notions of conformally invariant random metric in two dimensions. This is motivated by work of physicists in the 70s in the context of Liouville quantum gravity, and in particular by the so-called KPZ relation, which describes a way to relate geometric quantities associated with Euclidean models of statistical physics to their formulation in random geometry. I plan to discuss in an informal manner what is the problem and survey what is known rigorously. While the existence of a conformally invariant random metric is still open, I will explain a recent result of mine showing it is possible to construct a natural notion of Brownian motion in this geometry.