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Recent advances in mean-field theory are reviewed and applications to theSun, late-type stars, accretion disks, galaxies, and the early Universe arediscussed. We focus particularly on aspects of spatio-temporal nonlocality,which is one of the main insights that emerged from applying the test-fieldmethod to magnetic fields of different length and timescales. We also reviewthe status of nonlinear quenching and the relation to magnetic helicity, whichis an important observational diagnostic of modern solar dynamo theory. Bothsolar and some stellar dynamos seem to operate in an intermediate regime thathas not yet been possible to model successfully. This regime is bracketed byantisolar-like differential rotation on one end and stellar activity cyclesbelonging to the superactive stars on the other. The difficulty in modelingthis regime may be related to shortcomings in modelling solar/stellarconvection. On galactic and extragalactic length scales, the observationalconstraints are still less stringent and uncertain, but recent advances both intheory and in observations suggest that more conclusive comparisons may soon bepossible. The possibility of inversely cascading magnetic helicity throughoutall of the early Universe is particularly exciting in explaining the lowerlimits of magnetic fields on cosmological length scales and the possibility ofparity breaking and finite helicity of such a field.

Advances in mean-field dynamo theory and applications to astrophysical turbulence