Positive Displacement Machines are based on the concept of changing the volume available for fluid processing in a manner which produces matching variations in the pressure of that fluid. This is achieved by the use of mechanical linkages specially designed to optimize the pressure-volume relationships in order ofr the machine to perform efficiently. This volume explains the workings of a variety of positive displacement machines which include pumps, compressors, and gas expanders. Presented at a mathematical and technical level, the most influential research in this field over the past decade or so is examined, along with best practice from industry. The work featured here highlights the importance of using the latest computation techniques, and following proper design procedure to achieve the desired outcome.