These notes are based on a mini-course given during the conference Particle systems and PDE's -II which held at the Center of Mathematics of the University of Minho in December 2013. We discuss the problem of normal and anomalous diffusion of energy in systems of coupled oscillators perturbed by a stochastic noise conserving energy.The goal of statistical mechanics is to elucidate the relation between the microscopic world and the macroscopic world. Equilibrium statistical mechanics assume the microscopic systems studied to be in equilibrium. In this course we will be concerned with non-equilibrium statistical mechanics where time evolution is taken into account: our interest will not only be in the relation between the microscopic and the macroscopic scales in space but also in time.By microscopic system we refer to molecules or atoms governed by the classical Newton's equations of motion. The question is then to understand how do these particles manage to organize themselves in such a way as to form a coherent structure on a large scale. The "structure" will be described by few variables (temperature, pressure . . . ) governed by autonomous equations (Euler's equations, Navier-Stokes's equation, heat equation . . . ). The microscopic specificities of the system will appear on this scale only through the thermodynamics (equation of state) and