In this  course we  will investigate some questions  related to weak turbulence theory by using as explicit example of wave interactions the solutions to the 2D periodic cubic  Schrödinger equation. We will start by recalling  Strichartz estimates and well-posedness.  We will then explain how the evolution of the energy spectrum related to this equation can be studied in  two different ways. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms.  Then we will present some recent results on energy cascade via the analysis of radial  solutions to  the wave kinetic equation,  which is often used as the effective equation for the energy spectrum mentioned above.