Graph anomaly detection aims to find “strange” or “unusual” patterns in large graph or massive graph databases, and it has a wide range of application scenarios. Deep learning can learn the hidden rules from the data, and it has excellent performance in extracting potential complex patterns from data. With the great development of graph representation learning in recent years, how to detect graph anomaly using deep learning methods has attracted extensive attention in the area of academia and industry. Although a series of recent studies have investigated anomaly detection methods from the perspective of graphs, there is a lack of attention to graph anomaly detection methods under the background of deep learning. In this paper, we first give the definitions of various kinds of anomalies in static graph and dynamic graph and investigate the deep neural network based graph representation learning method and its various applications in graph anomaly detection. Then we present the current situation of research on graph anomaly detection based on deep learning from the perspective of static graph and dynamic graph, and summarize the application scenarios and related data sets of graph anomaly detection. At last, we discuss the current challenges and future research directions of graph anomaly detection.