The explanation of the observed galactic magnetic fields may require t
he existence of a primordial magnetic field. Such a field may arise du
ring the early cosmological phase transitions, or because of other par
ticle physics related phenomena in the very early universe reviewed he
re. The turbulent evolution of the initial, randomly fluctuating micro
scopic field to a large-scale macroscopic field can be described in te
rms of a shell model, which provides an approximation to the complete
magnetohydrodynamics. The results indicate that there is an inverse ca
scade of magnetic energy whereby the coherence of the magnetic field i
s increased by many orders of magnitude. Cosmological seed fields roug
hly of the order of 10(-20) G at the scale of protogalaxy, as required
by the dynamo explanation of galactic magnetic fields, thus seem plau
sible.