These is the text for my Harold Grad lecture, delivered in the 24th Rarefied Gas Dynamics conference, Bari (July 2004). They describe recent developments about convergence to equilibrium in kinetic theory, related to Boltzmann’s H
Theorem and entropy production. The style is intentionally informal, at the price of rigor and precision; full details can be found in the research papers quoted within the text. The text is partially based on my earlier contribution to the proceedings of the 14th International Congress of Mathematical Physics, Lisbon (July 2003).