This survey provides a comprehensive overview of state-of-the-art methods for anomaly detection in graph data. Anomaly detection is crucial in various domains such as security, finance, healthcare, and law enforcement. While traditional techniques focus on unstructured data, graph-based methods are increasingly important due to the prevalence of structured graph data. Graphs capture long-range correlations among interdependent data objects, making them suitable for detecting anomalies that are not easily identifiable in traditional data representations. The survey discusses various challenges in anomaly detection, including data-specific and problem-specific challenges. Data-specific challenges involve handling large, dynamic, and complex graph data. Problem-specific challenges include the lack of labeled data, class imbalance, and the need to detect novel anomalies. Graph-specific challenges include the relational nature of data, the diversity of anomaly definitions, and the large search space for complex anomalies. The survey categorizes anomaly detection methods into static and dynamic graphs. For static graphs, methods include structure-based and community-based approaches. Structure-based methods use graph features such as node degrees, centrality measures, and subgraph patterns. Community-based methods identify anomalies by detecting nodes or edges that connect different communities. For dynamic graphs, methods focus on detecting changes or events over time. The survey also emphasizes the importance of anomaly attribution and provides a framework for explaining detected anomalies. It highlights the need for effective and scalable methods that can handle large graphs and provide meaningful insights for further analysis. The survey concludes with a discussion of real-world applications of graph-based anomaly detection in various domains, including financial, auction, computer traffic, and social networks. The survey also addresses open theoretical and practical challenges in the field of graph-based anomaly detection.This survey provides a comprehensive overview of state-of-the-art methods for anomaly detection in graph data. Anomaly detection is crucial in various domains such as security, finance, healthcare, and law enforcement. While traditional techniques focus on unstructured data, graph-based methods are increasingly important due to the prevalence of structured graph data. Graphs capture long-range correlations among interdependent data objects, making them suitable for detecting anomalies that are not easily identifiable in traditional data representations. The survey discusses various challenges in anomaly detection, including data-specific and problem-specific challenges. Data-specific challenges involve handling large, dynamic, and complex graph data. Problem-specific challenges include the lack of labeled data, class imbalance, and the need to detect novel anomalies. Graph-specific challenges include the relational nature of data, the diversity of anomaly definitions, and the large search space for complex anomalies. The survey categorizes anomaly detection methods into static and dynamic graphs. For static graphs, methods include structure-based and community-based approaches. Structure-based methods use graph features such as node degrees, centrality measures, and subgraph patterns. Community-based methods identify anomalies by detecting nodes or edges that connect different communities. For dynamic graphs, methods focus on detecting changes or events over time. The survey also emphasizes the importance of anomaly attribution and provides a framework for explaining detected anomalies. It highlights the need for effective and scalable methods that can handle large graphs and provide meaningful insights for further analysis. The survey concludes with a discussion of real-world applications of graph-based anomaly detection in various domains, including financial, auction, computer traffic, and social networks. The survey also addresses open theoretical and practical challenges in the field of graph-based anomaly detection.