ABSTRACT: The modern search for a quantitative microscopic theory of macroscopic thermal phenomena dates to the middle of the nineteenth century. At that time the experiments of Joule and others made it clear that phenomena like boiling and freezing, heat conduction, diffusion, etc., have their origin in the dynamics of the atoms and molecules which are the constituents of matter. It was also soon recognized that the large disparity in the spatial and temporal scales between the world of atoms and the world of macroscopic experience not only necessitated a statistical theory but also assured that such a theory will give predictions precise enough to have the force of ``law'', as in Fourier's law or in the second law of thermodynamics. Thus was born the subject of statistical mechanics with Maxwell, Boltzmann, Gibbs and Einstein as its official parents (although of course it also had many grandparents, aunts, and uncles).

The twentieth century saw the development of the subject into a physically very successful and mathematically very beautiful theory of thermal equilibrium. The development of a comparable theory for the more complex world of nonequilibrium phenomena remains a challenge for the mathematical physicists of the twenty first century. A selective overview of where we stand at present in this endeavor will be the subject of my talk.