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Capturing high-dimensional (HD) data is a long-term challenge in signal processing and related fields. Snapshot compressive imaging (SCI) uses a 2D detector to capture HD (≥3D) data in a snapshot measurement. Via novel optical designs, the 2D detector samples the HD data in a compressive manner; following this, algorithms are employed to reconstruct the desired HD data cube. SCI has been used in hyperspectral imaging, video, holography, tomography, focal depth imaging, polarization imaging, microscopy, and so on. Although the hardware has been investigated for more than a decade, the theoretical guarantees have only recently been derived. Inspired by deep learning, various deep neural networks have also been developed to reconstruct the HD data cube in spectral SCI and video SCI. This article reviews recent advances in SCI hardware, theory, and algorithms, including both optimizationbased …