Uniformization of random surface and Liouville quantum gravity

In the last two decades there has been huge advance in understanding random surface from the perspective of the scaling limit of random triangulations. In the first part of the talk we review some highlights in this direction.  In particular, we will explain in what sense the so-called Liouville quantum gravity is the uniformization of a class of natural random surfaces, and report a joint work with Nina Holden rigorously proving that pure Liouville quantum gravity is the scaling limit of uniform triangulation under a conformal embedding related to Smirnov's proof of Cardy's formula. In the second part of the talk, we explain some ingredients of the proof, including the theory of mating of trees and dynamical percolation.