  Nonspherical mass motions are a generic feature of core-collapse supernovae,
and hydrodynamic instabilities play a crucial role for the explosion mechanism.
First successful neutrino-driven explosions could be obtained with
self-consistent, first-principle simulations in three spatial dimensions (3D).
But 3D models tend to be less prone to explosion than corresponding
axisymmetric (2D) ones. This has been explained by 3D turbulence leading to
energy cascading from large to small spatial scales, inversely to the 2D case,
thus disfavoring the growth of buoyant plumes on the largest scales. Unless the
inertia to explode simply reflects a lack of sufficient resolution in relevant
regions, it suggests that some important aspect may still be missing for robust
and sufficiently energetic neutrino-powered explosions. Such deficits could be
associated with progenitor properties like rotation, magnetic fields or
pre-collapse perturbations, or with microphysics that could lead to an
enhancement of neutrino heating behind the shock. 3D simulations have also
revealed new phenomena that are not present in 2D, for example spiral modes of
the standing accretion shock instability (SASI) and a stunning dipolar
lepton-emission self-sustained asymmetry (LESA). Both impose time- and
direction-dependent variations on the detectable neutrino signal. The
understanding of these effects and of their consequences is still in its
infancy.
