  Magnetic fields appear wherever plasma and currents can be found. As such,
they thread through all scales in Nature. It is natural, therefore, to suppose
that magnetic fields might have been formed within the high temperature
environments of the big bang. Such a primordial magnetic field (PMF) would be
expected to arise from and/or influence a variety of cosmological phenomena
such as inflation, cosmic phase transitions, big bang nucleosynthesis, the
cosmic microwave background (CMB) temperature and polarization anisotropies,
the cosmic gravity wave background, and the formation of large-scale structure.
In this review, we summarize the development of theoretical models for
analyzing the observational consequences of a PMF. We also summarize the
current state of the art in the search for observational evidence of a PMF. In
particular we review the framework needed to calculate the effects of a PMF
power spectrum on the CMB and the development of large scale structure. We
summarize the current constraints on the PMF amplitude $B_\lambda$ and the
power spectral index $n_B$ and discuss prospects for better determining these
quantities in the near future.
