Summary: Bizon and Rostworowski have recently suggested that anti-de Sitter spacetime might be nonlinearly unstable to transferring energy to smaller and smaller scales and eventually forming a small black hole. We consider pure gravity with a negative cosmological constant and find strong support for this idea. While one can start with certain linearized modes and add higher order corrections to construct nonlinear geons, this is not possible starting with a linear combination of two or more modes. One is forced to add higher frequency modes with growing amplitude. The implications of this turbulent instability for the dual-field theory are discussed.