We present the most relevant works on jet engine design that utilize thermodynamic cycle of detonative combustion. The efficiency advantages of thermodynamic detonative combustion cycle over Humphrey combustion cycle at constant volume and Brayton combustion cycle at constant pressure were demonstrated. An ideal Ficket-Jacobs detonation cycle, and the thermodynamic cycle of real detonation engine that utilizes over compressed detonation were discussed. Main trends in development of detonation engines were described. Relevant for nearest time problems and directions of research were formulated.