The talk will present recent progress in understanding the statistical physics of non-equilibrium diffusive systems. First, I will argue that by studying the probability of rare events in such systems one is able to probe the direct analog of a free energy in equilibrium systems. Methods for carrying out such calculations will be outlined. Finally, the analytical structure of the non-equilibrium free energy analogs, termed large deviation functionals, will be contrasted with that of equilibrium systems. In particular, it will be shown that 1) out of equilibrium the functionals are generically non-local functionals of the densities and 2) that in some models the functionals exhibit an unusual singular behaviour. The singularities are unique to non-equilibrium systems and are a direct consequence of the breaking of time-reversal symmetry.