  Isotropic homogeneous hydromagnetic turbulence is studied using numerical
simulations at resolutions of up to 1024^3 meshpoints. It is argued that, in
contrast to the kinematic regime, the nonlinear regime is characterized by a
spectral magnetic power that is decreasing with increasing wavenumber,
regardless of whether or not the turbulence has helicity. This means that the
root-mean-square field strength converges to a limiting value at large magnetic
Reynolds numbers. The total (magnetic and kinetic) energy spectrum tends to be
somewhat shallower than k^{-5/3}, in agreement with the findings of other
groups. In the presence of helicity, an inverse cascade develops, provided the
scale separation between the size of the computational box and the scale of the
energy carrying eddies exceeds a ratio of at least two. Finally, the
constraints imposed by magnetic helicity conservation on mean-field theory are
reviewed and new results of simulations are presented.
