This study utilizes a biomimetic autonomous underwater vehicle to perform moving-target tracking. A tracker with the structure of a fish-like design has advantages such as high energy efficiency, high maneuverability, and being quiet moving in water compared to conventional autonomous underwater vehicles. Kinematics models of the tracker vehicle, and the target vehicle are established. Measurement models of sensors on-board the tracker vehicle are provided. A Kalman Filter is applied to process measurements of the target and the tracker vehicle. The coordination of a pairs of pectoral fin and tail fin of the tracker is adopted to control the precise and agile motion of the tracker vehicle. A Doppler sonar is used to navigate the tracker and to calculate the position and directional angles of the target. A point sonar is used to detect and examine the relative distances of the target and the tracker in order to make a successful tracking. A series of successful tank experimental data are presented to validate the proposed tracking and control algorithms.