Abstract: The evolution, structure and origin of magnetic fields are stillopen 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 galaxy formation? I will explain how radio polarization surveys have the potential to address these questions. Specifically, I will present new Faraday rotation studies aimed at revealing the three-dimensional magnetic structure of the Milky Way, the Magellanic Clouds and distant galaxies. These studies pave the way to opening the full window to the magnetic Universe with the Square Kilometre Array and other future radio facilities.