The role of the electronic structure in determining the transport properties of ballistic point contacts is studied. The conductance in the ballistic regime is related to simple geometrical projections of the Fermi surface. The essential physics is first clarified for simple models. For real materials the band structure is taken into account using parameter-free local-spin-density approximation calculations. In magnetic metallic multilayers the electronic structure gives rise to a large difference in conductance between the parallel and antiparallel configurations. For Co/Cu and Fe/Cr multilayers the dependence of the conductances on the layer thicknesses and the crystal orientations is investigated for the geometries with the current perpendicular, parallel, and at an angle to the interface planes. In spite of the absence of spin-dependent scattering at defects, the ballistic giant magnetoresistance effects in the perpendicular geometry can be as large as 120% and 230% in Co/Cu and Fe/Cr multilayers, respectively. ͓S0163-1829͑98͒04612-8͔