Anomaly detection in hyperspectral images (HSIs) has attracted considerable interest in the remote-sensing domain, which aims to identify pixels with different spectral and spatial features from their surroundings. Most of the existing anomaly detection methods convert the 3-D data cube to a 2-D matrix composed of independent spectral vectors, which destroys the intrinsic spatial correlation between the pixels and their surrounding pixels, thus leading to considerable degradation in detection performance. In this article, we develop a tensor-based anomaly detection algorithm that can effectively preserve the spatial–spectral information of the original data. We first separate the 3-D HSI data into a background tensor and an anomaly tensor. Then the tensor nuclear norm based on the tensor singular value decomposition (SVD) is exploited to characterize the global low rank existing in both the spectral and spatial directions of the background tensor. In addition, the total variation (TV) regularization is incorporated due to the piecewise smoothness. For the anomaly component, the l2.1 norm is exploited to promote the group sparsity of anomalous pixels. In order to improve the ability of the algorithm to distinguish the anomaly from the background, we design a robust background dictionary. We first split the HSI data into local clusters by leveraging their spectral similarity and spatial distance. Then we develop a simple but effective way based on the SVD to select representative pixels as atoms. The constructed background dictionary can effectively represent the background materials and eliminate anomalies. Experimental results obtained using several real hyperspectral datasets demonstrate the superiority of the proposed method compared with some state-of-the-art anomaly detection algorithms.