The theory of "random surfaces" has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, and mathematical physics. I will give a friendly (I hope) colloquium-level overview of the subject with lots of pictures. Topics will include random planar maps (interpreted as discrete random surfaces), Liouville quantum gravity surfaces, conformal field theory. and the random fractal curves produced from the Schramm-Loewner evolution. Many of these topics are motivated by physics (statistical physics, string theory, quantum field theory, etc.) but they also have simple mathematical definitions that can be understood without a lot of physics background.