Abstract: The Jacobian variety of a smooth complex curve is a complex torus that admits two different algebraic descriptions. The Jacobian can be described as the Picard variety, which is the moduli space of line bundles, or it can be described as the Albanese variety, which is the universal abelian variety that contains the curve. I will talk about how to extend a family of Jacobians varieties by adding degenerate fibers. Corresponding to the two different descriptions of the Jacobian are two different extensions of the Jacobian: the Néron model, constructed by Néron, and the relative moduli space of stable sheaves, constructed by Langer, Maruyama, Simpson, and others. I will explain what these two extensions are and then prove that they are equivalent. This equivalence has surprising consequences for both the Néron model and the moduli space of stable sheaves, which I will explain.