In this paper, biomimetic underwater robots built using smart actuators, e.g., a shape memory alloy (SMA), an ionic polymer metal composite (IPMC), lead zirconate titanate (PZT), or a hybrid SMA and IPMC actuator, are reviewed. The effects of underwater environment were also considered because smart actuators are often affected by their external environment. The characteristics of smart actuators are described based on their actuating conditions and motion types. Underwater robots are classified based on different swimming modes. We expanded our classification to non-fish creatures based on their swimming modes. The five swimming modes are body/caudal actuation oscillatory (BCA-O), body/caudal actuation undulatory (BCA-U), median/paired actuation oscillatory (MPA-O), median/paired actuation undulatory (MPA-U), and jet propulsion (JET). The trends of biomimetic underwater robots were analyzed based on robot speed (body length per second, BL/s). For speed per body length, robots using an SMA as an actuator are faster than robots using an IPMC when considering a similar length or weight. Robots using a DC motor are longer while their speeds per body length are similar, which means that robots using smart actuators have an advantage of compactness. Finally, robots (using smart actuators or a motor) were compared with underwater animals according to their speed and different swimming modes. This review will help in setting guidelines for the development of future biomimetic underwater robots, especially those that use smart actuators.