The existence of magnetic fields is a mandatory requirement for the onset of most nonthermal phenomena in cosmological sources, especially gamma-ray burst sources and relativistic jet sources. The processes leading to the magnetization of the intergalactic medium are not yet known. Large-scale structures in the universe, like filaments and sheets of galaxies, evolve by the gravitational collapse of initially overdense regions giving rise to an intense relative motion of fully ionized gaseous matter and strong gaseous shock structures. We investigate analytically and numerically the generation of magnetic fields in the intergalactic medium by Weibel-type instabilities involving interpenetrating electron streams. Because of the hot temperatures of the intergalactic medium the investigation of the Weibel instability is based on the now available covariantly correct dispersion theory of linear waves, and thus improves on the existing non-relativistic treatments in the literature. These primordial Weibel magnetic fields may serve as cosmological seed fields for even stronger magnetic fields in cosmological sources.