PLEASE NOTE SPECIAL DAY (WEDNESDAY).   An important aspect of mathematical formulations of turbulence is to understand the propagation of randomness by the equations of fluid flow. My purpose in this talk is to describe the structure of the simplest rigorous model within this class: the evolution of random data by (nonlinear) scalar conservation laws. Our main discovery is that the evolution of the law of the solution is given by a completely integrable Hamiltonian system. This Hamiltonian system is described by kinetic equations presented as a Lax pair, has pleasing exact solutions, and has an algebraic structure that allows us to precisely link statistical hydrodynamics with a variety of integrable systems, old and new (e.g Manakov's top). Our work sits roughly at the intersection of kinetic theory, probability theory and integrable systems. Little background will be presumed in keeping with the spirit of a colloquium.