  The origin of magnetic fields in the Universe is an open problem in
astrophysics and fundamental physics. Polarization observations with the
forthcoming large radio telescopes will open a new era in the observation of
magnetic fields and should help to understand their origin. At low frequencies,
LOFAR (10-240 MHz) will allow us to map the structure of weak magnetic fields
in the outer regions and halos of galaxies, in galaxy clusters and in the Milky
Way via their synchrotron emission. Even weaker magnetic fields can be measured
at low frequencies with help of Faraday rotation measures. A detailed view of
the magnetic fields in the local Milky Way will be derived by Faraday rotation
measures from pulsars. First promising images with LOFAR have been obtained for
the Crab pulsar-wind nebula, the spiral galaxy M51, the radio galaxy M87 and
the galaxy clusters A2255 and A2256. With help of the polarimetric technique of
"Rotation Measure Synthesis", diffuse polarized emission has been detected from
a magnetic bubble in the local Milky Way. Polarized emission and rotation
measures were measured for more than 20 pulsars so far.
