  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 stable curve and in the case of
the universal family over the moduli space of stable pointed curves. In
particular, we construct new examples of fine compactified universal Jacobians.
On the other hand, we show that all fine compactified Jacobians of a single
stable curve of genus 1 can be obtained as moduli spaces of simple sheaves that
are stable with respect to some polarisation. Then we give a formula for the
Hodge and Betti numbers of each genus 1 fine compactified universal Jacobian
and prove that the even cohomology is algebraic.
