With the growth of social networks tremendous amount of data are generated in a regular interval of time that can be our search histories, likes, shares, posts, comments, etc. The generated data are publicly available which may become the prime target for the malicious users, who try to attack and harm the innocent users. Using anomaly detection techniques, we can identify the unusual behavior of such users. In social networks, anomalies can be detected by exploring the pattern hidden in the network. This paper mainly focuses on the graph mining techniques used for anomaly detection in social networks. The algorithm for anomaly detection using graph mining techniques has been categorized on the basis of different characteristics of anomalies, and the types of anomalies generated. In addition to this, the paper also emphasizes on the issues and challenges associated with each developed algorithm that will help the researchers and the scientists working in this area to find the solutions for problems associated with various algorithms.