  Let U be a pseudoconvex open set in a complex manifold M. When is U a Stein
manifold? There are classical counter examples due to Grauert, even when U has
real-analytic boundary or has strictly pseudoconvex points. We give new
criteria for the Steinness of U and we analyze the obstructions. The main tool
is the notion of Levi-currents. They are positive $\ddbar$-closed currents T of
bidimension (1,1) and of mass 1 directed by the directions where all continuous
psh functions in $U$ have vanishing Levi-form. The extremal ones, are supported
on the sets where all continuous psh functions are constant. We also construct
under geometric conditions, bounded strictly psh exhaustion functions, and
hence we obtain Donnelly- Fefferman weights. To any infinitesimally homogeneous
manifold, we associate a foliation. The dynamics of the foliation determines
the solution of the Levi-problem. Some of the results can be extended to the
context of pseudoconvexity with respect to a Pfaff-system.
