We will review the results of the ab initio study of the current-perpendicular-to-plane magnetotransport of a sample sandwiched by two ideal leads. The 'active' part of the system is either a trilayer consisting of two magnetic slabs of a finite thickness separated by a non-magnetic spacer, or a multilayer formed by alternating magnetic and non-magnetic layers. We use the Kubo-Landauer formulation of the conductance based on surface Green functions as formulated by means of the tight-binding linear muffin-tin orbital method. The formalism i s extended to the case of lateral supercells with random arrangements of atoms of two types, which in turn allows to deal with specular and diffusive scattering on equal footing. Applications refer to fcc-based Co/Cu/Co(001) multilayers including the transport through layers with non-collinear alignments of the magnetizations. We consider in detail the effect of substitutional alloying in the spacer as well as interdiffusion at the interfaces between magnetic and spacer layers.