Nonequilibrium statistical mechanics is a program rather than a theory. The program is to understand macroscopic properties of matter from microscopic dynamics. Here we take microscopic dynamics to be a perturbation of (classical) Hamiltonian dynamics. We are thus led to studying the ergodic theory of smooth dynamical systems to try to understand macroscopic non equilibrium physics. Fortunately we have now a fairly good understanding of the ergodic theory of some chaotic smooth dynamical systems, namely the results on hyperbolic systems by Liverani, Pollicott, Dolgopyat, Baladi, Tsujii, and many others. Unfortunately we know much less about non uniformly hyperbolic dynamics. The physics of transport phenomena, in particular heat transport, immediately leads to partially hyperbolic (non hyperbolic) dynamics; a difficult perturbation theorem by Dolgopyat barely provides some understanding of the situation. In the present lecture I shall discuss the problem of heat transport along the lines indicated above, and the mathematically related problem of the turbulent energy cascade in hydrodynamics.