  The aim of this paper is to study Seifert bundle structures on simply
connected 5--manifolds. We classify all such 5--manifolds which admit a Seifert
bundle structure, and in a few cases all Seifert bundle structures are also
classified. These results are then used to construct positive Ricci curvature
Einstein metrics on these manifolds.
  The proof has 4 main steps. First, the study of the Leray spectral sequence
of the Seifert bundle, based on work of Orlik--Wagreich. Second, the study of
log Del Pezzo surfaces. Third, the construction of K\"ahler--Einstein metrics
on Del Pezzo orbifolds using the algebraic existence criterion of
Demailly--Koll\'ar. Fourth, the lifting of the K\"ahler--Einstein metric on the
base of a Seifert bundle to an Einstein metric on the total space using the
Kobayashi--Boyer--Galicki method.
