  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.
