Algebraic curves are a central class of objects in algebraic geometry. Smooth curves in particular are a well-understood class of curves, and possess a robust moduli theory. On the other hand, curves with (nodal) singularities are somewhat less well-behaved; still, these curves feature prominently in modern algebraic geometry, in particular in the context of the Deligne-Mumford compactification of the moduli space of smooth curves. As such, it is of great interest to understand curves with these mild singularities. Over the past century, it became clear to algebraic geometers that much could be gained from studying combinatorial objects derived from these curves. In this talk, we survey this combinatorial approach and, if time permits, compare the construction of Jacobians in both the algebraic and combinatorial settings.