Non-equilibrium statistical mechanics, as opposed to its equilibrium counterpart, remains rather poorly understood. While equilibrium measures are easily characterized in terms of specific ensembles, the measures that characterize non-equilibrium states, whether stationary or time-dependent, are peculiar in that they are singular and display fractal properties. This observation has far-reaching consequences, in particular in connection to thermodynamics and the production of entropy. In this talk, Prof. Gilbert will use simple models based on random walks to review some of these properties, emphasizing the role of dynamics on the macroscopic transport processes they underlie. The lecture is directed to a general audience.