  We show that for each algebraic space that is separated and of finite type
over a field, and whose affinization is connected and reduced, there is a
universal morphism to a para-abelian variety. The latter are schemes that
acquire the structure of an abelian variety after some ground field extension.
This generalizes classical results of Serre on universal morphisms from
algebraic varieties to abelian varieties. Our proof relies on corresponding
facts for the proper case, together with the structural properties of group
schemes, removal of singularities by alterations, and ind-objects. It turns out
that the formation of the Albanese variety commutes with base-change up to
universal homeomorphisms. We also give a detailed analysis of Albanese maps for
algebraic curves and algebraic groups, with special emphasis on imperfect
ground fields.
