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The first report of a cosmic magnetic field outside the Earth was the result of a direct measurement of the Zeeman effect in the magnetic fields in sunspots in 1908. In 1950 it was suggested that the observed cosmic rays would require magnetic fields for their creation and their containment within the Galaxy. Optical polarization observations were first successful in 1949. Polarization of optical and infrared emission can also be caused by elongated dust grains which are aligned in magnetic fields due to the Davis-Greenstein mechanism first described in 1951. This interpretation was not accepted for a long time in the optical astronomy community. With the advent of radio astronomy this controversy was resolved and an active study of magnetic fields could begin.

Radio astronomy began in 1932 with the detection of continuum radio emission from the Milky Way. It became quickly clear that the observed radio waves were of a non-thermal nature and an interpretation of this phenomenon was actively sought. This was given in 1950 – the radio emission is due to relativistic cosmic-ray electrons gyrating in magnetic fields, emitting radio waves by the synchrotron process – when the theory of synchrotron emission theory was developed. In particular, it was soon pointed out that synchrotron emission should be highly polarized. In fact, in homogenous magnetic fields, up to 75% linear polarization of the continuum emission can be expected. This suggestion was taken up by observers of optical radiation who found in 1954 that the Crab Nebula was highly polarized and hence emitting light through the synchrotron process. The radio confirmation of the polarization of the Crab Nebula followed in 1957. The first definite detection of the linear polarization of the Galactic radio waves was published by in 1962. At the same time the polarization of the bright radio galaxy Cygnus A and the Faraday rotation of the polarization angles of the linearly polarized radio emission in Centaurus A were detected. Observations at two frequencies of a section of the Milky Way showed that the interstellar medium of the Milky Way can also cause Faraday effect. During this exciting time of definite detections of interstellar and extragalactic magnetic fields by observations of linear polarization, the Zeeman effect of radio spectral lines proved to be more elusive. Several groups attempted to measure magnetic fields by this direct method. It was in 1968 that finally the Zeeman effect at radio wavelengths was successfully observed in the absorption profile of the HI line in the direction of Cassiopeia A. From this time onward considerable data were collected on the distribution of magnetic fields in the Milky Way.

In the optical range, the polarization is produced by the different extinction along the minor and major axis of dust grains, while at far-infrared and submillimeter wavelengths the elongated dust grains themselves emit polarized emission, which was first detected in the 1980s. Progress has been slow, until recently an increase in reliable data became possible with the advent of submillimeter telescopes on excellent sites and sensitive polarimeters.

The first suggestions about the presence of magnetic fields in nearby galaxies were made in 1958, based on observations of the polarization of stars in the Andromeda galaxy, M31. In 1967 observations of the linear polarization of diffuse starlight started in bright nearby galaxies. In 1970 the polarization of stars in the Magellanic Clouds implied the presence of magnetic fields in these neighboring galaxies. Low-frequency radio observations of galaxies showed non-thermal spectra and hence indicated the presence of magnetic fields. The first detection of the linear polarization of the radio emission from nearby galaxies in 1972 led the way to massive improvement on our knowledge of the morphology of magnetic fields in galaxies. These early radio observations were in good agreement with the early optical polarization studies of galaxies.

In this review, the status of our knowledge about the magnetic fields in our Milky Way and in nearby star-forming galaxies is summarized. Magnetic fields are a major agent in the interstellar and intra- cluster medium and affect the physical processes in various ways. They contribute significantly to the total pressure which balances the gas disk of a galaxy against gravitation. Magnetic reconnection is a possible heating source for the interstellar medium (ISM) and halo gas. They affect the dynamics of the turbulent ISM and the gas flows in spiral arms. The shock strength in spiral density waves is decreased and structure formation is reduced in the presence of a strong field. The interstellar fields are closely connected to gas clouds. Magnetic fields stabilize gas clouds and reduce the star-formation efficiency to the observed low values. On the other hand, magnetic fields are essential for the onset of star formation as they enable the removal of angular momentum from protostellar clouds via ambipolar diffusion. MHD turbulence distributes energy from supernova explosions within the ISM and drives field amplification and ordering via the dynamo mechanism. In galaxies with low star-formation activity or in the outer disks, the magneto-rotational instability can generate turbulence and heat the gas. Magnetic fields control the density and distribution of cosmic rays in the ISM. Cosmic rays accelerated in supernova remnants can provide the pressure to drive a galactic outflow and generate buoyant loops of magnetic fields (through the Parker instability). Understanding the interaction between the gas and the magnetic field is a key to understand the physics of galaxy disks and halos and the evolution of galaxies.

The magnetic field of the Milky Way is of particular importance for experiments to detect ultrahigh-energy cosmic rays (UHECRs). Results from the first years of AUGER indicate that the arrival directions of detected UHECRs with energies of more than 1019 eV show a statistically significant coincidence with the positions of known nearby active galaxies. This interpretation only holds if the deflections in the magnetic fields of the intergalactic medium and the Milky Way halo are not larger than a few degrees. However, little is known about the structure and strength of the magnetic field in the halo of our Milky Way and beyond.

There is one class of galaxies where magnetic fields play a crucial role: "active" galaxies which are governed by a central Black Hole. The formation of jets and radio lobes can only be understood with the presence of magnetic fields. The physics of these phenomena is quite different from that in "normal" star-forming galaxies and will not be discussed in this review.

Magnetic fields have also been detected in the intergalactic medium surrounding the galaxies in a cluster through observations of non-thermal diffuse radio halos and the Faraday effect of background radio sources seen through the cluster. These intracluster magnetic fields are probably generated by turbulent gas motions as the result of massive interactions between galaxies and the intracluster gas. Magnetic fields affect thermal conduction in galaxy clusters and hence their evolution. Outflows from galaxies may have magnetized the intergalactic medium, so that the general intergalactic space may be pervaded with magnetic fields. Unfortunately, cosmic rays and dust grains are missing outside of galaxies and galaxy cluster, and magnetic fields remain invisible. Intracluster magnetic fields are also beyond the scope of this review.

Cosmological models of structure formation indicate that the intergalactic space is probably permeated by magnetic filaments. Galactic winds, jets from active galaxies and interactions between galaxies can magnetize the intergalactic medium. The detection of magnetic fields in intergalactic filaments and observations of the interaction between galaxies and the intergalactic space is one of the important tasks for future radio telescopes. Until now the arguments for the presence of magnetic fields in the distant Universe is based on observations of the non-thermal radio emission and Faraday rotation in galaxies at high redshift. Magnetic fields existed already in QSOs at epochs with redshifts of at least z 5 and in starburst galaxies at redshifts of at least z ≈ 4, but the earliest magnetic fields are yet to be discovered (section 5).

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