Underwater acoustic sensor networks (UASNs), which are formed by a number of interconnected mobile vehicles and static sensors, have emerged as a promising solution to explore and utilize the ocean resources. Typically, the localization, communication, and control are the fundamental services for the applications of UASNs. Although they are closely related, the localization, communication, and control issues are usually separately tackled. The separate design directly affects the localization accuracy, transmission reliability, and control efficiency, especially for resource-constrained UASNs. In this regard, it is essential and necessary to co-design the localization, communication, and control systems for UASNs. Presently, the theoretical framework of the above integration design is still in the construction phase, and some key problems remain unresolved. Therefore, this article aims to give a survey on the integration design of localization, communication, and control for UASNs. We first present the communication architecture, through which the main challenges aiming at the integration design are analyzed. After that, a holistic survey on the underwater localization, communication, and control basics is provided. Followed by this, the recent advances on the integration design are given. Finally, we make an outlook to the future research directions on the integration design for UASNs.