Compactified Jacobians of singular curves

The Jacobian variety of a smooth curve is an Abelian variety that carries important informations about the curve itself. Its properties have been widely studied along the decades, giving rise to a significant amount of beautiful mathematics.
However, for singular curves, the situation is more involved since the generalized Jacobian variety is not anymore an Abelian variety, once it is, in general, not compact.
The problem of compactifying it is, of course, very natural, and it is considered to go back to the work of Igusa and Mayer-Mumford
in the 50's-60's.
Since then, several solutions appeared, differing from one another in various aspects as the generality of the construction, the modular description of the boundary and the functorial properties.
In this talk I will start by recalling some of these constructions and how they relate to each other. I will then report on several new results, partially obtained in collaboration with Filippo Viviani, which aim to understand better the geometry of these moduli spaces as well as some generalizations of these constructions.