Picking a surface, uniformly at random

What does a random surface look like? And what does it mean anyway to pick a surface uniformly at random?

These questions and related others arise naturally in the study of Liouville quantum gravity, and have been the subject of intense research on the mathematical side in the last few years. I will try to give a panoramic view of a canonical notion of random surface, including counter-intuitive aspects of its geometry. I will also explain some of the outstanding conjectures in this field, as well as some progress that has been made in particular concerning Liouville Brownian motion, a natural notion of diffusion in this geometry.