Abstract: The present paper aims to provide a summary report on recent research progress about motion characteristics and flow mechanism of a shock train in a hypersonic inlet-isolator with complex background waves to help the researchers working on hypersonic inlet-isolator easily with their further work. It covers shock train motion characteristics, mechanism of a shock train jumps, and the method for the model of a shock train jumps with complex background waves. At first, the investigations for the motion characteristics of shock train with fixed or variable background waves are described, which point out that the boundary layer separation caused by the alternating favorable or adverse pressure gradient in the isolator is the physical mechanism of the shock train motion characteristics. Followed, the triggering mechanism and condition of shock train jumps in an isolator with background waves are discussed. At last, based on the understanding of the motion characteristics and jump mechanism, the method for the mathematical model of the shock train motion in an isolator with background waves is given to provide a reference for the control of the shock train leading edge.