  We show that the Jacobians of prestable curves over toroidal varieties always
admit N\'eron models. These models are rarely quasi-compact or separated, but
we also give a complete classification of quasi-compact separated group-models
of such Jacobians. In particular we show the existence of a maximal
quasi-compact separated group model, which we call the saturated model, which
has the extension property for all torsion sections. The N\'eron model and the
saturated model coincide over a Dedekind base, so the saturated model gives an
alternative generalisation of the classical notion of N\'eron models to
higher-dimensional bases; in the general case we give necessary and sufficient
conditions for the N\'eron model and saturated model to coincide. The key
result, from which most others descend, is that the logarithmic Jacobian of
\cite{Molcho2018The-logarithmic} is a log Neron model of the Jacobian.
