How can you sample a surface uniformly at random? A natural approach is to consider a uniformly sampled planar map, which is a model for a discrete surface studied in many branches of both math and physics. When the size of the surface goes to infinity it converges to the continuum random surface known as a Liouville quantum gravity surface, which was originally introduced in the physics literature. We will give an introduction to these objects and present a powerful technique to study them known as conformal welding, where the random fractal curves known as Schramm-Loewner evolutions appear.