Anomaly detection is to identify data that is significantly different from other normal patterns in the data set, and is widely applied in fraud detection, intrusion detection, and data analysis and other fields. Existing researches on anomaly detection are mostly based on unstructured data point sets, and there are complex structural relationships between data to form a complex network in the real world, and the network is represented as a graph in mathematical form, so the demand of anomaly detection for complex networks is increasing. This paper summarizes the current methods and research advances of anomaly detection for complex networks. First, the necessity and development history of anomaly detection for complex networks are proposed. Then, from the perspective of static and dynamic graphs, the anomaly detection for complex networks is divided into static graph anomaly detection based on structure, community, relationship learning, and dynamic graph anomaly detection based on nodes, edges, subgraphs, and full graphs, and then summarize, analyze and compare by category, and the application scenarios of anomaly detection for complex networks are given. Finally, the future research directions of anomaly detection for complex networks are summarized.