Moduli of generalized elliptic curves
Moduli spaces of elliptic curves with various kinds of level structure are fundamental objects in arithmetic geometry. Unfortunately, they are not proper, so it is natural to seek modular compactifications by allowing certain degenerations of elliptic curves. Deligne and Rapoport showed that by considering moduli spaces of so-called generalized elliptic curves, one obtains compactifications of modular curves. I will talk about what generalized elliptic curves are and what some of their properties are, and sketch a proof that moduli spaces of generalized elliptic curves exist.