As a cross-cutting field between ocean development and multi-robot system (MRS), the underwater multi-robot system (UMRS) has gained increasing attention from researchers and engineers in recent decades. In this paper, we present a comprehensive survey of cooperation issues, one of the key components of UMRS, from the perspective of the emergence of new functions. More specifically, we categorize the cooperation in terms of task-space, motion-space, measurement-space, as well as their combination. Further, we analyze the architecture of UMRS from three aspects, i.e., the performance of the individual underwater robot, the new functions of underwater robots, and the technical approaches of MRS. To conclude, we have discussed related promising directions for future research. This survey provides valuable insight into the reasonable utilization of UMRS to attain diverse underwater tasks in complex ocean application scenarios.