The theory of ``random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, combinatorics, analysis and mathematical physics. Just as Brownian motion is a special kind of random path, there is a similarly special kind of random surface, which is characterized by special symmetries, and which arises in many different contexts. Random surfaces are often motivated by physics: statistical physics, string theory, quantum field theory, and so forth. They have also been independently studied by mathematicians working in random matrix theory and enumerative graph theory. But even without that motivation, one may be drawn to wonder what a ``typical" two-dimensional manifold look likes, or how one can make sense of that question. I will give a broad overview of what this theory is about, including many computer simulations and illustrations. In particular, I will highlight some recent work with Jason Miller in which we proved the equivalence of Liouville quantum gravity and the Brownian sphere --- two random surface models that were historically defined in completely different ways.