The talk begins with a review of classical black hole dynamics in four dimensions, and the thermodynamic analogy that led Hawking to his discovery of the thermal particle emission from black holes, and the Hawking Temperature. It is explained how the classical laws of black hole dynamics become the first and second laws of thermodynamics. The situation becomes more subtle when the black holes live in a background spacetime with a cosmological constant. Understanding the thermodynamics in this case is important for the AdS/CFT Correspondence, which relates bulk properties in the spacetime to those of a field theory on the boundary. Physically, the most important case is for a five-dimensional bulk. All previous results for the thermodynamics of rotating AdS black holes in more than four dimensions are incorrect, and violate the first law of thermodynamics. The results presented in this talk rectify this situation.