The circumstances under which negative absolute temperatures can occur are discussed, and principles of thermodynamics and statistical mechanics at negative temperatures are developed. If the entropy of a thermodynamic system is not a monotonically increasing function of its internal energy, it possesses a negative temperature whenever (∂S∂U)X is negative. Negative temperatures are hotter than positive temperatures. When account is taken of the possibility of negative temperatures, various modifications of conventional thermodynamics statements are required. For example, heat can be extracted from a negative-temperature reservoir with no other effect than the performance of an equivalent amount of work. One of the standard formulations of the second law of thermodynamics must be altered to the following: It is impossible to construct an engine that will operate in a closed cycle and provide no effect other than (1) the extraction of heat from a positive-temperature reservoir with the performance of an equivalent amount of work or (2) the rejection of heat into a negative-temperature reservoir with the corresponding work being done on the engine. A thermodynamic system that is in internal thermodynamic equilibrium, that is otherwise essentially isolated, and that has an energetic upper limit to its allowed states can possess a negative temperature. The statistical mechanics of such a system are discussed and the results are applied to nuclear spin systems.