  Magnetism is one of the four fundamental forces. However, the origin of
magnetic fields in stars, galaxies and clusters is an open problem in
astrophysics and fundamental physics. When and how were the first fields
generated? Are present-day magnetic fields a result of dynamo action, or do
they represent persistent primordial magnetism? What role do magnetic fields
play in turbulence, cosmic ray acceleration and galaxy formation? Here we
demonstrate how the Square Kilometer Array (SKA) can deliver new data which
will directly address these currently unanswered issues. Much of what we
present is based on an all-sky survey of rotation measures, in which Faraday
rotation towards >10^7 background sources will provide a dense grid for probing
magnetism in the Milky Way, in nearby galaxies, and in distant galaxies,
clusters and protogalaxies. Using these data, we can map out the evolution of
magnetised structures from redshifts z>3 to the present, can distinguish
between different origins for seed magnetic fields in galaxies, and can develop
a detailed model of the magnetic field geometry of the intergalactic medium and
of the overall Universe. With the unprecedented capabilities of the SKA, the
window to the Magnetic Universe can finally be opened.
