Compared with traditional underwater vehicles, bio-inspired fish robots have the advantages of high efficiency, high maneuverability, low noise, and minor fluid disturbance. Therefore, they have gained an increasing research interest, which has led to a great deal of remarkable progress theoretically and practically in recent years. In this review, we first highlight our enhanced scientific understanding of bio-inspired propulsion and sensing underwater and then present the research progress and performance characteristics of different bio-inspired robot fish, classified by the propulsion method. Like the natural fish species they imitate, different types of bionic fish have different morphological structures and distinctive hydrodynamic properties. In addition, we select two pioneering directions about soft robotic control and multi-phase robotics. The hybrid dynamic control of soft robotic systems combines the accuracy of model-based control and the efficiency of model-free control, and is considered the proper way to optimize the classical control model with the intersection of multiple machine learning algorithms. Multi-phase robots provide a broader scope of application compared to ordinary bionic robot fish, with the ability of operating in air or on land outside the fluid. By introducing recent progress in related fields, we summarize the advantages and challenges of soft robotic control and multi-phase robotics, guiding the further development of bionic aquatic robots.