As mankind steps into the 21st century, the exploration of the ocean also enters a new phase. In the recent decades, the research on biomimetic robot fish has attracted scientific communities' attention increasingly. With its character of high flexibility and intelligence, biomimetic robot fish have become one of the research focuses in the field of underwater robots in recent years. This paper mainly describes the classification of propulsion modes and the corresponding characteristics, the international research situation, the open problems and future research issues for biomimetic robot fish.