Boltzmann's 1872 H-theorem was a glorious failure: the Loschmidt-Culverwell (reversibility) and Zermelo (recurrence) objections clearly showed something was wrong with it, but it led to the birth of statistical mechanics. This talk concentrates on historical and conceptual aspects of the theorem and its discontents. On the historical side, the radical nature of the critique by Poincaré and Zermelo is exposed, and Boltzmann's response to it is discussed. On the conceptual side, the nature of the theorem is examined and the relative strengths of the main objections assessed. Several old and modern misconceptions about the H-theorem are clarified.