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With the continuous development of aerospace technology and remote sensing technology, the applications of remote sensing images in lots of fields have been expanding. Hyperspectral image (HSI) is a common type of remote sensing image which consists of a series of two-dimensional remote sensing images as a 3D data cube. Each two-dimensional image in HSI can reflect the reflection/radiation intensity of different wavelengths of electromagnetic waves, and each pixel of HSI corresponds to the spectral curve reflecting the spectral information in different wavelengths. Therefore, the hyperspectral remote sensing images have the feature of "map-spectrum integration", which contains not only the spectral information with strong discriminative, but also rich spatial information, so the hyperspectral data have great application potential. Hyperspectral anomaly detection is to detect the pixels in a scene with different characteristics from surrounding pixels and determines them as anomalous targets without the any prior knowledge of the target. Since hyperspectral anomaly detection is an unsupervised process that does not require any priori information about the target to be measured in advance, it plays a great role in real life. For example, anomaly target detection technology can be used to search and rescue people after a disaster, to quickly determine the fire point of a forest fire, and to search mineral point in mineral resource exploration. Hyperspectral anomaly detection has been a popular research direction in the area of remote sensing image processing in recent years and a large number of researchers have conducted extensive and in-depth research and achieved rich research results. However, hyperspectral anomaly detection still faces many difficult problems, such as the targets of the same material may exhibit different spectral characteristics due to the different imaging equipment and imaging environment, which may interfere with the detection results and lead to the problem of "same object with different spectrum", and the targets of different materials may also exhibit the problem of "different object with different spectrum". Then, most of the existing hyperspectral anomaly detection algorithms are only in the laboratory stage and with low technology maturity. Besides, the hyperspectral data may have lots of spectral bands that contains a large amount of redundant information, which makes the data processing difficult. Moreover, the number of publicly available hyperspectral anomaly detection datasets is insufficient and most of the datasets are very old. In this paper, we firstly summarize the main research progress of hyperspectral anomaly detection, and then classify and summarize the existing mainstream algorithms, mainly divided into five categories: statistics-based anomaly detection methods, data expression-based anomaly detection methods, data decomposition-based methods anomaly detection, deep learning-based methods anomaly detection and other methods. Besides, through the investigation, analysis and summary of the existing methods, three future development directions of hyperspectral anomaly detection are proposed, including database expansion: introduce newer dataset with more images and more sophisticated remote sensing sensors, multi-source data combination: take advantages of different imaging sensors and different types of remote sensing data; algorithm practical: enables the anomaly detection algorithms to be ported for application on real platforms.