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We describe the magnetic and transport properties of Fe(OOI )/Cr(OOI) superlattices grown on GaAs (001) by molecular-beam epitaxy and characterized by reflection high-energy electron diffraction (RHEED), Auger spectroscopy, x-ray diffraction, and electron microscopy. For Cr layers thinner than about 30 A the magnetic behavior reveals strong antiferromagnetic couplings between the Fe layers across the Cr layers. Polarized neutron diffraction experiments confirm the existence of an antiferromagnetic superstructure. We discuss the origin of the antiferromagnetic (AF) coupling. The Fe/Cr superlattices with AF interlayer coupling exhibit a giant magnetoresistance: when an applied field aligns the magnetizations of the Fe layers, the resistivity drops by a factor of 2 for some samples. This giant magnetoresistance can be ascribed to the spin dependence of the electron scattering by interfaces. We compare our results with the predictions of two recent theoretical models.