  We present results of a detailed and fully non-linear numerical and
analytical investigation of magnetic field evolution from the very earliest
cosmic epochs to the present. We find that, under reasonable assumptions
concerning the efficiency of a putative magnetogenesis era during cosmic phase
transitions, surprisingly strong magnetic fields 10^{-13} - 10^{-11} Gauss, on
comparatively small scales 100 pc - 10 kpc may survive to the present. Building
on prior work on the evolution of magnetic fields during the course of
gravitational collapse of a cluster, which indicates that pre-collapse fields
of 4\times 10^{-12} Gauss extant on small scales may suffice to produce
clusters with acceptable Faraday rotation measures, we question the widely hold
view that cluster magnetic fields may not be entirely of primordial origin.
