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 physics of quantum gravity, and have been the subject of intense research on the mathematical side in the last few years.
After discussing aspects of a canonical notion of random surface, I will discuss interesting and 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.