  Blazar observations point toward the possible presence of magnetic fields
over intergalactic scales of the order of up to $\sim1\,$Mpc, with strengths of
at least $\sim10^{-16}\,$G. Understanding the origin of these large-scale
magnetic fields is a challenge for modern astrophysics. Here we discuss the
cosmological scenario, focussing on the following questions: (i) How and when
was this magnetic field generated? (ii) How does it evolve during the expansion
of the universe? (iii) Are the amplitude and statistical properties of this
field such that they can explain the strengths and correlation lengths of
observed magnetic fields? We also discuss the possibility of observing
primordial turbulence through direct detection of stochastic gravitational
waves in the mHz range accessible to LISA.
