We have studied the giant magnetoresistance ͑GMR͒ in magnetic multilayer point contacts of three different types. The first generation contacts were made by deposition with molecular-beam epitaxy ͑MBE͒ of an uncoupled Co/Cu multilayer on a pre-etched hole in a thin membrane. These devices exhibited a GMR, but its ratio was low and, as deduced from finite element calculations, in many cases was dominated by the resistance of the multilayer electrode. When corrected for this, the maximum point-contact GMR was 3%. The multilayer structure at some depth in the constriction was disrupted, as observed by transmission electron microscopy. This was identified as a cause of the low GMR, together with contamination and an oxide layer in the constriction, resulting from ex situ sample rotation. The second generation was fabricated by sputtering of a coupled Co/Cu multilayer before etching of the nanohole, giving a proper multilayer at the constriction. Further, the GMR signal from the electrode was shorted by a thick Cu cap. This did not bring the expected increase of the GMR (ratioр5%), indicating that the so-called dead layers and the quality of the interface between the GMR system and the contacting metal were limiting. This interface quality was strongly improved for the third generation of contacts by using in situ rotation, while the question of multilayer quality was avoided by shifting to granular Co/Au. Granular Co/Au in the constriction was obtained by growing a discontinuous Co layer by MBE. The maximum GMR ratio of the granular contacts was 14%, an improvement of a factor 3. These contacts displayed small jumps in the GMR, two-level fluctuations in the resistance time trace and ballistic transport, the latter being evident from phonon peaks in the point-contact spectrum of a high resistance contact. ͓S0163-1829͑99͒01537-4͔