The topics reported in the present review suggest various open problems on the nature of large-scale magnetic fields in the present and in the early Universe:
in principle the morphology of magnetic fields in spiral galaxies could be used in order to understand the origin of galactic magnetism but in practice the observations offer answers which are often contradictory;
recent observations of magnetic fields inside Abell clusters confirm the presence of strong magnetic fields in the µ G range;
there is evidence that also superclusters are magnetized;
in spite of the fact that magnetic fields may modify the patterns of the CMB anisotropies and, eventually, induce anisotropies in the polarization, the theoretical analysis has been rarely corroborated by the comparison with the available experimental data;
Faraday rotation of the CMB polarization would be a powerful tool to study the possible exsitence of large scale magnetic fields prior to recombination;
the imprints of magnetic fields on the relic background of gravitational waves (in the window of wide band interferometers) allows to test the existence of a background of Abelian gauge fields at the electroweak epoch.
It would be important, for the theorist, a reasonable accuracy (within one order of magnitude) in the experimental determination of large-scale magnetic fields. For instance even one order of magnitude in the intensity of the intra-cluster magnetic field (i.e. 10-6 G rather than 10-7 G) makes a difference as far as the problem of the origin is concerned. In the same perspective, more accurate determinations of the typical correlation scales of the intra-cluster fields would be desirable.
There is, at the moment, no compelling reason why large-scale magnetic fields should not be primordial, at least to some degree:
in the early Universe magnetic fields are easily produced during phase transitions but their typical correlation scales are small;
inflation greatly helps in producing correlations over very large scales and the (early) variation of gauge couplings during the inflationary phase allows the generation of intense large-scale fields;
primordial magnetic fields may have an impact on the thermodynamical history of the Universe, but, in practice, the obtained constraints improve only marginally, in various interesting cases, the critical density bound imposed on the magnetic energy density;
if magnetic fields are generated during phase transitions, a growth in their correlation scale cannot be excluded because of turbulent phenomena whose existence can be reasonably expected (but not firmly predicted) on the basis of our terrestrial knowledge of magnetized plasmas.
In the next decade a progress of the empirical knowledge is expected in apparently unrelated areas like x and -ray astronomy, radio-astronomy, CMB physics, detection of relic gravitational waves. All these areas are connected with the existence and with the properties of large-scale magnetic fields in the early and in the present Universe. This connection is sometimes rather fragile since it is mediated by various theoretical assumptions. However, there is the hope that, in a not too distant future, some of the puzzles related to the origin and existence of large-scale magnetic fields may be resolved and some of the current theoretical guessworks firmly ruled out.
Some of the ideas presented in this review have been elaborated and assembled on the occasion of different sets of lectures delivered through the years. The author wishes to express his gratitude to D. Babusci, H. de Vega, M. Gasperini, H. Kurki-Suonio, E. Keihänen, N. Sanchez, M. Shaposhnikov, G. Veneziano, A. Vilenkin. A special thank is due to G. Cocconi for important remarks which improved the first draft.