  The goal of this work is to pursue the study of pseudo-effective line bundles
and vector bundles. Our first result is a generalization of the Hard Lefschetz
theorem for cohomology with values in a pseudo-effective line bundle. The
Lefschetz map is shown to be surjective when (and, in general, only when) the
pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This
result has several geometric applications, e.g. to the study of compact
K\"ahler manifolds with pseudo-effective canonical or anti-canonical line
bundles. Another concern is to understand pseudo-effectivity in more algebraic
terms. In this direction, we introduce the concept of an "almost" nef line
bundle, and mean by this that the degree of the bundle is nonnegative on
sufficiently generic curves. It can be shown that pseudo-effective line bundles
are almost nef, and our hope is that the converse also holds true. This can be
checked in some cases, e.g. for the canonical bundle of a projective 3-fold.
  From this, we derive some geometric properties of the Albanese map of compact
K\"ahler 3-folds.
