  In this paper we give a detailed account of the first 3D relativistic
magnetohydrodynamic (MHD) simulations of Pulsar Wind Nebulae (PWN), with
parameters most suitable for the Crab Nebula. In order to clarify the new
features specific to 3D models, reference 2D simulations have been carried out
as well. Compared to the previous 2D simulations, we considered pulsar winds
with much stronger magnetisation, up to \sigma=3, and accounted more accurately
for the anticipated magnetic dissipation in the striped zone of these winds.
While the 3D models preserve the separation of the post termination shock flow
into the equatorial and polar components, their relative strength and
significance differ. Whereas the highly magnetised 2D models produce highly
coherent and well collimated polar jets capable of efficient "drilling" through
the supernova shell, in the corresponding 3D models the jets are disrupted by
the kink mode current driven instability and "dissolve" into the main body of
PWN after propagation of several shock radii. Our results show that contrary to
the expectations based on 1D analytical and semi-analytical models, our
numerical solutions with highly magnetized pulsar winds still produce
termination shocks with radii comparable to those deduced from the
observations. We present polarized synchrotron maps constructed from the 3D
simulations, showing that the wealth of morphological features observed with
Hubble and Chandra can well be reproduced with 3D-MHD models.
