Engines and methods execute a high efficiency hybrid cycle, which is implemented in a volume within an engine. The cycle includes isochoric heat addition and over-expansion of the volume within the engine, wherein the volume is reduced in a compression portion of the cycle from a first quantity to a second quantity, the volume is held substantially constant at the second quantity during a heat addition portion of the cycle, and the volume is increased in an expansion portion of the cycle to a third quantity, the third quantity being larger than the first quantity.