The celebrated Kolmogorov’s K41 theory of fully developed turbulence attempts to explain and quantify “wild, but typical” behaviors of “chaotic” fluids, most notably the lack of conservation of the total energy. The loss of energy is not due to friction between fluid molecules, but rather to the limited regularity of the flow. Kolmogorov’s theory is numerically and experimentally validated to a very large extent, however, very little is known in rigorous mathematical terms. In my lecture, I will present some aspects of Kolmogorov’s theory, and illustrate recent results on a related question for the linear advection equation.