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Anomaly detection from graph data is an important data mining task in many applications such as social networks, finance, and e-commerce. Existing efforts in graph anomaly detection typically only consider the information in a single scale (view), thus inevitably limiting their capability in capturing anomalous patterns in complex graph data. To address this limitation, we propose a novel framework, graph ANomaly dEtection framework with Multi-scale cONtrastive lEarning (ANEMONE in short). By using a graph neural network as a backbone to encode the information from multiple graph scales (views), we learn better representation for nodes in a graph. In maximizing the agreements between instances at both the patch and context levels concurrently, we estimate the anomaly score of each node with a statistical anomaly estimator according to the degree of agreement from multiple perspectives. To further exploit a handful of ground-truth anomalies (few-shot anomalies) that may be collected in real-life applications, we further propose an extended algorithm, ANEMONE-FS, to integrate valuable information in our method. We conduct extensive experiments under purely unsupervised settings and few-shot anomaly detection settings, and we demonstrate that the proposed method ANEMONE and its variant ANEMONE-FS consistently outperform state-of-the-art algorithms on six benchmark datasets.