This talk will motivate and briefly describe the construction of a weakly turbulent solution of the cubic nonlinear Schrodinger equation on the two dimensional torus. In particular, the solution will start off small in a high Sobolev norm and eventually the solution will grow larger than any target threshold in the same norm. This talk reports on joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao.