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For a variety V, its Albenese variety 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 generallyCan we say something similar for an arbitrary scheme? 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?
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"Albanese" schemes: When does an "initial abelian scheme" exist under a given scheme?

For a variety V, its Albenese variety 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?