The Boltzmann equation is the central equation of kinetic theory, and it describes the evolution of the phase space density for a dilute gas. Boltzmann's famous H-theorem says that the entropy is strictly monotone increasing for solutions of this equation unless the solution is in equilibrium.  For many years discussion of this result centered on understanding how  such irreversibility could arise from reversible Newtonian dynamics, and important questions  in this direction remain open. However, more recent work has sought a quantitative version of the H-theorem than can be used to quantify the rate of approach to equilibrium. The first results in this direction were obtained by myself and C. Carvalho, and were further developed and sharpened by Toscani and Villani, and they figure in the work for which Villani won the Fields medal. The problem of relating entropy to entropy production has been found to be applicable to many problems beside the Boltzmann equation, and has suggested a number of interesting functional inequalities that are the subject of recent works, and it has also suggested several open problems. These will be explained here, assuming, of course, no prior knowledge of kinetic theory.