The automatic identification system (AIS) was introduced in the maritime domain to increase the safety of sea traffic. AIS messages are transmitted as broadcasts to nearby ships and contain, among others, information about the identification, position, speed, and course of the sending vessels. AIS can thus serve as a tool to avoid collisions and increase onboard situational awareness. In recent years, AIS has been utilized in more and more applications since it enables worldwide surveillance of virtually any larger vessel and has the potential to greatly support vessel traffic services and collision risk assessment. Anomalies in AIS tracks can indicate events that are relevant in terms of safety and also security. With a plethora of accessible AIS data nowadays, there is a growing need for the automatic detection of anomalous AIS data. In this paper, we survey 44 research articles on anomaly detection of maritime AIS tracks. We identify the tackled AIS anomaly types, assess their potential use cases, and closely examine the landscape of recent AIS anomaly research as well as their limitations.