Motivated by Kolmogorov's theory of hydrodynamic turbulence, we considerdissipative weak solutions to the 3D incompressible Euler equations and the 2D surface quasi-geostrophic equations. We prove that up to a certain regularity threshold weak solutions are not unique. In the case of the Euler system this is the threshold determined by the Onsager conjecture. For SQG, this answers an open problem posed by De Lellis and Szekelyhidi Jr.