Abstract: A novel thermodynamic engines including a piston operating on a compressible fluid in a thermally insulated volume, which also includes a movable displacer which selectively divides the internal volume between a warm and a cold side, and a regenerator through which the fluid from the selectively divided volume passes and transfer its heat to or receives heat from, wherein the piston and displacer are each periodically moved in various complex motions according to the present invention to provide efficiency higher than Carnot efficiency. The resulting novel structures and methods, generally referred to as “Superclassical Cycle” engines, incorporate constant volume cooling and aspects of the “Proell Effect” (relative to cooling) to achieve improved efficiencies wherein the gas temperature on the cold side of a fluid displacer is below the lowest regenerator temperature due to “self-refrigeration.