The origin and evolution of cosmic magnetic fields, their strength and structure in intergalactic space, their first occurrence in young galaxies, and their dynamical importance for galaxy evolution remain widely unknown. Radio synchrotron emission, its polarization and its Faraday rotation are powerful tools to study the strength and structure of magnetic fields in galaxies. Unpolarized radio synchrotron emission traces isotropic turbulent fields which are strongest in spiral arms and bars (20–30 μG) and in central starburst regions (50–100 μG). Such fields are dynamically important; they can affect gas flows and drive gas inflows in central regions. Polarized radio emission traces ordered fields which can be regular or anisotropic turbulent, generated from isotropic turbulent fields by compression or shear. The strongest ordered fields of 10–15 μG strength are generally found in interarm regions and follow the orientation of adjacent gas spiral arms. In galaxies with strong density waves, ordered (anisotropic turbulent) fields are also observed at the inner edges of the spiral arms. Ordered fields with spiral patterns exist in grand-design, barred and flocculent galaxies, and in central regions of starburst galaxies. Ordered fields in interacting galaxies have asymmetric distributions and are an excellent tracer of past interactions between galaxies or with the intergalactic medium. Irregular galaxies host isotropic turbulent fields often of similar strength as in spiral galaxies, but only weak ordered fields. Faraday rotation measures (RM) of the diffuse polarized radio emission from the disks of several galaxies reveal large-scale spiral patterns that can be described by the superposition of azimuthal modes; these are signatures of regular fields generated by a mean-field α −Ω dynamo. So far no indications were found in external galaxies of large-scale field reversals, like the one in the Milky Way. Ordered magnetic fields are also observed in radio halos around edge-on galaxies out to large distances from the plane, with X-shaped patterns. In the outflow cone above a starburst region of NGC 253, RM data indicate a helical magnetic field.