This feature article is a written version of a lecture that Dan Margalit, professor in the School of Mathematics, gave at the 2022 Joint Mathematical Meetings of the American Mathematical Society (AMS). The Society established the Maryam Mirzakhani Lecture in 2018 to honor the memory of Mirzakhani, the first woman and first Iranian to win the Fields Medal, one of the highest honors in math. Margalit writes that on a basic level, Mirzakhani’s work centers around the geometry of surfaces, as understood through their simple curves: "Starting from this humble-seeming topic, Mirzakhani made surprising and sweeping connections between numerous fields of mathematics, including algebraic geometry, Teichmüller theory, moduli spaces, dynamics, homogeneous spaces, symplectic geometry, and billiards."