  Let X be a smooth proper variety over a perfect field k of arbitrary
characteristic. Let D be an effective divisor on X with multiplicity. We
introduce an Albanese variety Alb(X, D) of X of modulus D as a higher
dimensional analogon of the generalized Jacobian of Rosenlicht-Serre with
modulus for smooth proper curves. Basing on duality of 1-motives with unipotent
part (which are introduced here), we obtain explicit and functorial
descriptions of these generalized Albanese varieties and their dual functors.
  We define a relative Chow group of zero cycles w.r.t. the modulus D and show
that Alb(X, D) is a universal quotient of this Chow group.
  As an application we can rephrase Lang's class field theory of function
fields of varieties over finite fields in explicit terms.
