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Traditional spectral-based methods such as PCA are popular for anomaly detection in a variety of problems and domains. However, if data includes tensor (multiway) structure (e.g. space-time-measurements), some meaningful anomalies may remain invisible with these methods. Although tensor-based anomaly detection (TAD) has been applied within a variety of disciplines over the last twenty years, it is not yet recognized as a formal category in anomaly detection. This survey aims to highlight the potential of tensor-based techniques as a novel approach for detection and identification of abnormalities and failures. We survey the interdisciplinary works in which TAD is reported and characterize the learning strategies, methods and applications; extract the important open issues in TAD and provide the corresponding existing solutions according to the state-of-the-art.