We give, for a complex algebraic variety S, a Hodge realization functor F Hdg S from the (un-bounded) derived category of constructible motives DAc(S) over S to the (undounded) derived category D(M HM (S)) of algebraic mixed Hodge modules over S. Moreover, for f : T → S a morphism of complex quasi-projective algebraic varieties, F Hdg − commutes with the four operations f * , f * , f ! , f ! on DAc(−) and D(M HM (−)), making in particular the Hodge realization functor a morphism of 2-functor on the category of complex quasi-projective algebraic varieties which for a given S sends DAc(S) to D(M HM (S)), moreover F Hdg S commutes with tensor product. We also give an algebraic and analytic Gauss-Manin realization functor from which we obtain a base change theorem for algebraic De Rham cohomology and for all smooth morphisms a relative version of the comparaison theorem of Grothendieck between the algebraic De Rahm cohomology and the analytic De Rahm cohomology.