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.
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