For a variety V, its [Albenese variety][1] Alb(V) is a variety with a map V → Alb(V) which factors uniquely into any map from V to an abelian variety.  How about more generally?  When do we know there exists an "Albanese" scheme Alb(X)?  That is,

>Under what conditions on a scheme X does there exist a morphism X → Alb(X) which factors uniquely into any map from from X to an abelian scheme?


  [1]: http://en.wikipedia.org/wiki/Albanese_variety