Anti-de Sitter (AdS) spacetime is linearly stable, but non-linearly unstable to a weak gravitational turbulent instability. Regardless of how weak the initial scalar or gravitational perturbation of AdS is, this instability forces the system to transfer energy from low to high frequency modes. This energy cascade is similar to what happens with the familiar process of turbulence. In a full time evolution, the scalar instability leads to the formation of a black hole and a similar endpoint is conjectured for the turbulent gravitational instability. In the context of the gravity/gauge theory correspondence, this gravitational instability of AdS provides a holographic description of quantum turbulence in the dual field theory.