Capturing high-dimensional (HD) data is a long-term challenge in signal processing and related fields. Snapshot compressive imaging (SCI) uses a two-dimensional (2D) detector to capture HD ($ge3$D) data in a {em snapshot} measurement. Via novel optical designs, the 2D detector samples the HD data in a {em 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, etc.~Though 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 optimization-based and deep-learning-based algorithms. Diverse applications and the outlook of SCI are also discussed.
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