Universal compactified Jacobians

In this paper we describe compactifications of the universal Jacobian stack of line bundles over smooth curves obtained by considering open substacks of the moduli stack of torsion-free rank-1 simple sheaves on reduced curves of given genus. The results we describe are mostly expository and based on Altman-Kleiman and Esteves constructions of compactified Jacobians for families of reduced curves. For the moduli stack of Deligne-Mumford n-pointed stable curves, we study fine compactifications obtained by imposing stability conditions on the sheaves and we prove that in this situation we get Deligne-Mumford stacks endowed with a projective coarse moduli space.