This paper summarizes a talk given in the PDE Session at the 2006 International Congress on Mathematical Physics about joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao. We build new smooth solutions of the cubic defocussing nonlinear Schrödinger equation on the two dimensional torus which are weakly turbulent: given any δ 1, K 1, s > 1, we construct smooth initial data u 0 in the Sobolev space H s with u 0 H s < δ, so that the corresponding time evolution u satisfies u(T) H s > K at some time T .