Underwater wireless optical communications are an emerging solution to the expanding demand for broadband links in oceans and seas. In this paper, a cellular underwater wireless optical code division multiple-access network is proposed to provide broadband links for commercial and military applications. The optical orthogonal codes are employed as signature codes of underwater mobile users. Fundamental key aspects of the network, such as its backhaul architecture, its potential applications, and its design challenges, are presented. In particular, a promising underwater localization and positioning scheme based on this cellular network is presented. Furthermore, the proposed network is used as infrastructure of centralized, decentralized and relay-assisted underwater sensor networks for high-speed real-time monitoring. Finally, probable design challenges, such as cell edge coverage, blockage avoidance, power control, and network capacity, are addressed.