The explanation of the observed galactic magnetic fields may require the existence of a primordial magnetic field. Such a field may arise during the early cosmological phase transitions, or because of other particle physics related phenomena in the very early universe reviewed here. The turbulent evolution of the initial, randomly fluctuating microscopic field to a large-scale macroscopic field can be described in terms of a shell model, which provides an approximation to the complete magnetohydrodynamics. The results indicate that there is an inverse cascade of magnetic energy whereby the coherence of the magnetic field is increased by many orders of magnitude. Cosmological seed fields roughly of the order of 10-20 G at the scale of protogalaxy, as required by the dynamo explanation of galactic magnetic fields, thus seem plausible.