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Let $X$ be a proper smooth variety over a field $k$ of characteristic $0$ and $Y$ an effective divisor on $X$ with multiplicity. We introduce a generalized Albanese variety Alb$(X,Y)$ of $X$ of modulus $Y$, as higher dimensional analogue of the generalized Jacobian with modulus of Rosenlicht-Serre. Our construction is algebraic. For $k=ℂ$ we give a Hodge theoretic description.