The present study focuses on summarizing the recent advancements in the field of fish swimming mode research and bionic robotic fish prototypes based on smart materials. It has been widely acknowledged that fish exhibit exceptional swimming efficiency and manoeuvrability compared to conventional underwater vehicles. In the pursuit of developing autonomous underwater vehicles (AUVs), conventional experimental methods often prove to be complex and expensive. Hence, the utilization of computer simulations for hydrodynamic modelling provides a cost-effective and efficient approach for analysing the swimming behaviour of bionic robotic fish. Additionally, computer simulations can provide data that are difficult to obtain through experimental methods. Smart materials, which integrate perception, drive, and control functions, are increasingly being applied to bionic robotic fish research. However, the utilization of smart materials in this field is still an area of ongoing research and several challenges remain unresolved. This study provides an overview of the current state of research on fish swimming modes and the development of hydrodynamic modelling. The application of four distinct types of smart materials in bionic robotic fish is then reviewed, with a focus on analysing the advantages and disadvantages of each material in driving swimming behaviour. In conclusion, the paper highlights the key technical challenges that must be addressed for the practical implementation of bionic robotic fish and provides insights into the potential future directions of this field.