  In this article we study the topological structure of the lifts to the
universal of the stable and unstable foliations of $3$-dimensional Anosov
flows. In particular we consider the case when these foliations do not have
Hausdorff leaf space. We completely determine the structure of the set of non
separated leaves from a given leaf in one of these foliations. As a consequence
of this suspensions are characterized, up to topological conjugacy, as the only
$3$-dimensional Anosov flows without freely homotopic closed orbits.
Furthermore the structure of branching is related to the topology of the
manifold: if there are infinitely many leaves not separated from each other,
then there is an incompressible torus transverse to the flow. Transitivity is
not assumed for these results. Finally, if the manifold has negatively curved
fundamental group we derive some important properties of the limit sets of
leaves in the universal cover.
