I will begin my talk wiht a brief overview of weak turbulence where I will try to give an intuitive picture of the phenomenon as well as a few concrete physical examples of it. Indeed, I will refer back to the example of water waves to illustrate much of the talk. Next, I will try to give a "welcoming" introductionn to the calculations of weak turbulence. Equipped with the results of these calculations, I will discuss the relationship between weak turublence, Kolmogorov spectra and intermittency.