Synchrotron emission, its polarization and its Faraday rotation at radio frequencies of 0.2-10 GHz are powerful tools to study the strength and structure of cosmic magnetic fields. Unpolarized emission traces turbulent fields which are strongest in galactic spiral arms and bars (20-30 mu G) and in central starburst regions (50-100 mu G). Such fields are dynamically important, e. g. they can drive gas inflows in central regions. Polarized emission traces ordered fields which can be regular (uni-directional) or anisotropic random (generated from isotropic random fields by compression or shear). Ordered fields with spiral patterns exist in grand-design, barred and flocculent galaxies, and in central regions of starburst galaxies. The strongest ordered (mostly regular) fields of 10-15 mu G strength are generally found in galactic interarm regions and follow the orientation of adjacent gas spiral arms. Faraday rotation measures (RM) of the diffuse polarized radio emission from the disks of several spiral galaxies reveal large-scale patterns, which are signatures of regular fields probably generated by a mean-field dynamo. Ordered fields in interacting galaxies have asymmetric distributions and are an excellent tracer of past interactions between galaxies or with the intergalactic medium. Ordered magnetic fields are also observed in radio halos around edge-on galaxies, out to large distances from the plane, with X-shaped patterns. - The strength of the total magnetic field in our Milky Way is about 6 mu G near the solar radius, but several mG in dense clouds, pulsar wind nebulae, and filaments near the Galactic Center. Diffuse polarized radio emission and Faraday rotation data from pulsars and background sources show spiral fields with large-scale reversals, but the overall field structure in our Galaxy is still under debate. - Diffuse radio emission from the halos of galaxy clusters is mostly unpolarized because intracluster magnetic fields are turbulent, while cluster "relics", probably shock fronts by cluster mergers, can have degrees of polarization of up to 60% and extents of up to 2 Mpc. The IGM magnetic field strength is >= 3 10(-16) G with a filling factor of at least 60%, derived from the combination of data from the HESS and FERMI telescopes. - Polarization observations with the forthcoming large radio telescopes will open a new era in the observation of cosmic magnetic fields and will help to understand their origin. At low frequencies, LOFAR (10-250 MHz) will allow us to map the structure of weak magnetic fields in the outer regions and halos of galaxies and galaxy clusters. Small Faraday rotation measures can also be best measured at low frequencies. Polarization at higher frequencies (1-10 GHz), as observed with the EVLA, MeerKAT, APERTIF and the SKA, will trace magnetic fields in the disks and central regions of nearby galaxies in unprecedented detail. The SKA pulsar survey will find many new pulsars; their RMs will map the Milky Way's magnetic field with high precision. All-sky surveys of Faraday rotation measures towards a dense grid of polarized background sources with the SKA and its precursor telescope ASKAP are dedicated to measure magnetic fields in distant intervening galaxies, galaxy clusters and intergalactic filaments, and will be used to model the overall structure and strength of the magnetic field in the Milky Way. With the SKA, ordered fields in distant galaxies and cluster relics can be measured to redshifts of z similar or equal to 0. 5, turbulent fields in starburst galaxies or cluster halos to z similar or equal to 3 and regular fields in intervening galaxies towards QSOs to z similar or equal to 5.
