  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.
