In the talk presented by one of us (AF) at Cargese, the mechanisms of the giant magnetoresistance (MR) in magnetic multilayers were discussed very generally for both the CIP (current in plane) and CPP (current perpendicular to the planes) geometries. A so general discussion would be too long for these proceedings and the present paper will be restricted to the CPP case. We present the theoretical model that we have recently worked out [1,2] and discuss its application to experimental data obtained for the Ag/Co and Cu/Co systems at Michigan State University [3,4,5] and presented at Cargese by Pr. P.A. Schroeder [6]. We describe the specific fundamental problems related to the spin accumulation effects occuring in the CPP geometry and we calculate the magnetoresistance. The expressions of the MR become relatively simple in the limit where the layer thicknesses are much smaller than the spin diffusion length and we justify the analysis of experimental results developed at Michigan State University [3–6]. We also relate our theory to those of Johnson et al [7,8], van Son et al [9] and Zhang and Levy [10]. Of course, it is not in the scope of the present paper to develop calculations presented elsewhere in detail [1] and we will focus on the presentation of the basis of the model and the discussion of our results.