The space of line bundles of fixed degree on a smooth compact curve is an abelian variety. On a singular curve, however, line bundles can degenerate to sheaves whose rank jumps at the singularities. I will give an introduction to the moduli of such objects, called compactified Jacobians, and explain how to think of them as degenerate abelian varieties. I will show how, in the case of nodal curves, they admit a beautiful description in terms of toric geometry.