This paper extensively reviews hybrid rocket propulsion-related activities from combustion engine designs to launch tests. Starting with a brief review of rocket propulsion development history, a comparison among the three bi-propellant rocket propulsion approaches, and hybrid rocket engine design guidelines, a very thorough review related to hybrid rocket propulsion and its applications is presented in this paper. In addition to propellant choice, engine design also affects the hybrid rocket performance and, therefore, a variety of engine designs, considering, e.g., fuel geometry, swirl injection, ignition designs, and some innovative flow-channel designs are also explored. Furthermore, many fundamental studies on increasing hybrid rocket engine performances, such as regression rate enhancement, mixing enhancement, and combustion optimization, are also reviewed. Many problems that will be encountered for practical applications are also reviewed and discussed, including the O/F ratio shift, low-frequency instability, and scale-up methods. For hybrid rocket engine applications in the future, advanced capabilities and lightweight design of the hybrid rocket engine, such as throttling capability, thrust vectoring control concept, insulation materials, 3D-printing manufacturing technologies, and flight demonstrations, are also included. Finally, some active hybrid rocket research teams and their plans for flight activities are briefly introduced.