Magnetic fields are a fundamental part of many astrophysical phenomena, but the evolution, structure and origin of magnetic fields are still unresolved problems in physics and astrophysics. When and how were the first fields generated? Are present-day magnetic fields the result of standard dynamo action, or do they represent rapid or recent field amplification through other processes? What role do magnetic fields play in turbulence, cosmic ray acceleration and structure formation? I explain how the Square Kilometre Array (SKA), a next-generation radio telescope, can deliver stunning new data-sets that will address these currently unanswered issues. The foundation for these experiments will be an all-sky survey of rotation measures, in which Faraday rotation toward > 10 7 background sources will provide a dense grid for probing magnetism in the Milky Way, nearby galaxies, and in distant galaxies, clusters and protogalaxies. Using these data, we can map out the evolution of magnetized 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. In addition, the SKA will certainly discover new magnetic phenomena beyond what we can currently predict or imagine.