  We introduce a general abstract notion of fine compactified Jacobian for
nodal curves of arbitrary genus.
  We focus on genus 1 and prove combinatorial classification results for fine
compactified Jacobians in the case of a single nodal curve and in the case of
the universal family over the moduli space of stable pointed curves. We show
that if the fine compactified Jacobian of a nodal curve of genus 1 can be
extended to a smoothing of the curve, then it can be described as the moduli
space of stable sheaves with respect to some polarisation. In the universal
case we construct new examples of genus 1 fine compactified universal
Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus
1 fine compactified universal Jacobian and prove that their even cohomology is
algebraic.
