  In this paper we describe compactified universal Jacobians, i.e.
compactifications of the moduli space of line bundles on smooth curves obtained
as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an
approach due to Oda-Seshadri. We focus on the combinatorics of the stability
conditions used to define compactified universal Jacobians. We explicitly
describe an affine space, the stability space, with a decomposition into
polytopes such that each polytope corresponds to a proper Deligne-Mumford stack
that compactifies the moduli space of line bundles. We apply this description
to describe the set of isomorphism classes of compactified universal Jacobians
(answering a question of Melo), and to resolve the indeterminacy of the
Abel-Jacobi sections (addressing a problem raised by Grushevsky-Zakharov).
