In this talk we review analytical and numerical studies of hydrodynamic vortices in conformal fluids and their gravity duals. We present two conclusions. First, (3+1)-dimensional turbulence is within the range of validity of the AdS/hydrodynamics correspondence. Second, the local equilibrium of the fluid is equivalent to the ultralocality of the holographic correspondence, in the sense that the bulk data at a given point is determined, to any given precision, by the boundary data at a single point together with a fixed number of derivatives. With this criterion we see that the cores of hot and slow (3+1)-dimensional conformal generalizations of Burgers vortices are everywhere in local equilibrium and their gravity duals are thus easily found. On the other hand local equilibrium breaks down in the core of singular (2+1)-dimensional vortices, but the holographic correspondence with Einstein gravity may be used to define the boundary field theory in the region in which the hydrodynamic description fails.