Aquatic robotics is making a critical transition to adapt and inspire more efficient systems from nature. The result is the abandonment of inefficient propeller-based locomotion for a biological locomotion type suitable for the specific mission. Bioinspired aquatic unmanned vehicles (AUVs) could be exploited in a diverse range of missions depending on the design and its capabilities. Removing the human pilot and creating an animal-based AUV means that more hazardous aquatic environments can be studied with reduced repercussions. There is a diverse range of biological locomotion's to choose from when developing a bioinspired AUV. The respective animals that exhibit a specific swimming mode give a range of criteria to follow make the system more capable of swimming. In this review, how previous AUV developers determined the kinematic, physical, and hydrodynamic modeling of these systems are consolidated and discussed. All types of developed actuators are reviewed, organized, and explained based on their materials and motion capabilities. The electronic components of these systems are outlined to give an idea of how these bioinspired AUVs are constructed. Then, it is discussed what makes these systems bioinspired and biomimetic, and the trends of these designs. Limitations and future recommendations on possible improvements for these systems are offered and deeply discussed.