An autonomous vessel uses multiple different radio technologies such as satellites, mobile networks and dedicated narrowband systems, to connect to other ships, services, and the remote operations center (ROC). In-ship communication is mainly implemented with wired technologies but also wireless links can be used. In this survey paper, we provide a short overview of autonomous and remote-controlled systems. This paper reviews 5G-related standardization in the maritime domain, covering main use cases and both the role of autonomous ships and that of people onboard. We discuss the concept of a connectivity manager, an intelligent entity that manages complex set of technologies, integrating satellite and terrestrial technologies together, ensuring robust in-ship connections and ship-to-outside connections in any environment. This survey paper describes the architecture and functionalities of connectivity management required for an autonomous ship to be able to operate globally. As a specific case example, we have implemented a research environment consisting of ship simulators with connectivity components. Our simulation results on the effects of delays to collision avoidance confirm the role of reliable connectivity for safety. Finally, we outline future research directions for autonomous ship connectivity research, providing ideas for further work.