The intersection theory of the moduli space of stable curves is one of the central topics of enumerative geometry, a subject with connections to Gromov-Witten theory, integrable systems, and complex geometry among others. One of the most fruitful ways to study a curve is through its interaction with its Jacobian. The intersection theory of the universal Jacobian is however much less developed than that of curves, especially over the locus of singular curves -- which is at least partially due to the subtleties involved in compactifying spaces of line bundles of singular curves. In this talk, I will explain how ideas from logarithmic geometry allow us to study the intersection theory of compactified Jacobians systematically, leading to new results both about their geometry and the enumerative geometry of curves.