I will give a emph{very} introductory, informal talk—no prerequisites beyond a first course in algebraic geometry—on work using moduli spaces of curves to explore general questions in birational geometry over the past 50 years. The first half of the talk will recall basic notions from birational geometry and then review what moduli spaces are, sticking to the case of curves. The second half will outline classical constructions of these spaces and illustrate the interplay between constructions of a moduli space $M$, the birational geometry of $M$ and the geometry of curves that $M$ classifies, especially in families.