Autonomous underwater vehicles (AUVs) are submersible underwater vehicles controlled by onboard computers. AUV formation is a cooperative control which focuses on controlling multiple AUVs to move in a group while executing tasks. In contrast to a single AUV, multi-AUV formation represents higher efficiency and better stability for many applications, such as oil and gas industries, hydrographic surveys, and military missions, etc. To achieve better formation, there are several key factors, including AUV performance, formation control, and communication capability. However, most studies in the field of AUV formation mainly focus on formation control methods. We observe that the research of communication capability and AUV performance of multiple AUV formation is still in an early stage. It is beneficial to researchers to present a comprehensive survey of the state of the art of AUV formation research and development. In this paper, we study AUV, formation control, and underwater acoustic communication capability in detail. We propose a classification framework with three dimensions, including AUV performance, formation control, and communication capability. This framework provides a comprehensive classification method for future AUV formation research. It also can be used to compare different methods and help engineers choose suitable formation methods for various applications. Moreover, our survey discusses formation architecture with communication constraints and we identify some common misconceptions and questionable research for formation control related to communication.