In her thesis, Mirzakhani counted the number of simple closed geodesics of bounded length on a hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, homogeneous dynamics, symplectic geometry, and surface topology. In this talk, I’ll discuss some of these connections and a qualitative strengthening of her theorem, describing what these curves, and their complements, actually (generically) look like. This is joint work with Francisco Arana-Herrera.