  We study complex projective manifolds X that admit surjective endomorphisms
f:X->X of degree at least two. In case f is etale, we prove structure theorems
that describe X. In particular, a rather detailed description is given if X is
a uniruled threefold. As to the ramified case, we first prove a general theorem
stating that the vector bundle associated to a Galois covering of projective
manifolds is ample (resp. nef) under very mild conditions. This is applied to
the study of ramified endomorphisms of Fano manifolds with second Betti number
one. It is conjectured that the projective space is the only Fano manifold
admitting admitting an endomorphism of degree d>1, and we prove that in several
cases.
  A part of the argumentation is based on a new characterization of the
projective space as the only manifold that admits an ample subsheaf in its
tangent bundle.
