The purpose of this paper is to study the notion of freeness for reduced complete intersections, which is a generalization of the notion of free divisors introduced by K. Saito. We give some properties of multi-logarithmic differential forms and their multi-residues along a reduced complete intersection defined by a regular sequence (h_1,. .. , h_k). We first establish a kind of duality between multi-logarithmic differential k-forms and multi-logarithmic k-vector fields. We use it to prove a duality between the Jacobian ideal and multi-residues. The main result is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic forms. We then focus on quasi-homogeneous curves, for which we compute explicitly a minimal free resolution of the module of multi-logarithmic forms.