  In the last decade, the relativistic magnetohydrodynamic (MHD) modelling of
pulsar wind nebulae, and of the Crab nebula in particular, has been highly
successful, with many of the observed dynamical and emission properties
reproduced down to the finest detail. Here, we critically discuss the results
of some of the most recent studies: namely the investigation of the origin of
the radio emitting particles and the quest for the acceleration sites of
particles of different energies along the termination shock, by using wisps
motion as a diagnostic tool; the study of the magnetic dissipation process in
high magnetization nebulae by means of new long-term three-dimensional
simulations of the pulsar wind nebula evolution; the investigation of the
relativistic tearing instability in thinning current sheets, leading to fast
reconnection events that might be at the origin of the Crab nebula gamma-ray
flares.
