Abstract: Observations show that galaxies have magnetic fields with a component that is coherent over a large fraction of the galaxy with field strength of order microGauss and coherence scale of order 10 kpc. Understanding the origin of these fields is one of the challenging questions of modern astrophysics. There are two scenarios under current discussion: an astrophysical one, where the needed seed field is generated on smaller scales, and a cosmological scenario where the seed field is generated prior to galaxy formation in the early universe on scales that are large at the present epoch. In this talk I will address the following questions: (i) How and when was this magnetic field generated? (ii) How does it evolve during the expansion of the universe? (iii) Can the amplitude and statistical properties of the magnetic field explain the strengths and correlation lengths of the magnetic fields observed in galaxies and clusters? (iv) Is the seed magnetic field detectable through cosmological observations, such as CMB anisotropy measurements or LSS observations? And (v) if yes, what are the observational constraints on such a primordial magnetic field?