Maritime safety and security are gaining increasing concern in recent years. There are a growing number of studies aiming at improving situational awareness in the maritime domain by identifying vessel anomaly behaviors based on the data provided by the Automatic Identification System (AIS). Two types of data-driven methods are most popular in vessel anomaly detection based on AIS data: the statistical methods and machine learning methods. To improve the detection model efficiency and accuracy, hybrid models are formed by combining different types of methods. In order to incorporate expert knowledge, interactive systems are also designed and realized. In this paper, we provide a review of the popular statistical and machine learning models, as well as the hybrid models and interactive systems based on the data-driven methods used for anomaly detection based on AIS data.