  Recent advances in mean-field theory are reviewed and applications to the
Sun, late-type stars, accretion disks, galaxies, and the early Universe are
discussed. We focus particularly on aspects of spatio-temporal nonlocality,
which is one of the main insights that emerged from applying the test-field
method to magnetic fields of different length and timescales. We also review
the status of nonlinear quenching and the relation to magnetic helicity, which
is an important observational diagnostic of modern solar dynamo theory. Both
solar and some stellar dynamos seem to operate in an intermediate regime that
has not yet been possible to model successfully. This regime is bracketed by
antisolar-like differential rotation on one end and stellar activity cycles
belonging to the superactive stars on the other. The difficulty in modeling
this regime may be related to shortcomings in modelling solar/stellar
convection. On galactic and extragalactic length scales, the observational
constraints are still less stringent and uncertain, but recent advances both in
theory and in observations suggest that more conclusive comparisons may soon be
possible. The possibility of inversely cascading magnetic helicity throughout
all of the early Universe is particularly exciting in explaining the lower
limits of magnetic fields on cosmological length scales and the possibility of
parity breaking and finite helicity of such a field.
