  In this paper we classify noetherian hereditary abelian categories satisfying
Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a
classification of saturated noetherian hereditary categories.
  As a side result we show that when our hereditary categories have no nonzero
projectives or injectives, then the Serre duality property is equivalent to the
existence of almost split sequences.
