It is an effective way to execute a complicated mission by cooperating unmanned vehicles. This paper focuses on a search- and-track (SAT) mission for an underwater target, and the mission is implemented by combining an unmanned aerial vehicle (UAV), an unmanned surface vehicle (USV) and an autonomous underwater vehicle (AUV). In the cooperative path planning model, the mission is divided into the search phase and the track phase, and the goals of the two phases are to maximize the search space and minimize the terminal error respectively. The constraints contain the maneuverability of vehicles and communication ranges between vehicles. Strategies based on random simulation experiments and asynchronous planning are developed to design the cooperative path planning algorithm in the two phases, and the paths are generated by an improved particle swarm optimization (IPSO) algorithm in a centralized or a distributed mode. Simulation results demonstrate that the proposed method can deal with different situations. The UAV&USV&AUV system is superior to the USV&AUV system in the SAT mission.