How many circles are tangent to 3 fixed circles? I'll explain how moduli spaces, a central topic in algebraic geometry, help shed light on the answer to this question and others like it. We'll start by discussing the moduli space of circles. This is an example of a moduli space of "embedded curves." Next, I'll explain the distinction between an "embedded curve" and its associated "abstract curve" (also called a "Riemann surface"). I'll finish by sharing some recent results about moduli spaces of abstract curves, which are joint work with Samir Canning.