The paper provides a comprehensive overview of snapshot spectral imaging (SSI) technology, focusing on its system architecture, mathematical modeling, and recent advancements. SSI enables the capture of complete spectral information of objects in a very short time, making it suitable for dynamic observations in various fields such as environmental monitoring, medical diagnostics, and industrial inspection. The review highlights the evolution from early spatial-spectral mapping methods to more advanced techniques that encode and reconstruct various dimensions of light, including amplitude, phase, and wavelength. Metasurfaces, which offer unprecedented control over optical properties at sub-wavelength scales, have become a key component in SSI, reducing system size and improving efficiency. The paper discusses the integration of metasurfaces in SSI, detailing their optical principles, mathematical models, and modulation devices. It also explores different data acquisition schemes and detector capture modes, categorizing techniques into non-computational spatial-spectral mapping and computationally required coded reconstruction methods. The review covers various types of SSI systems, including integral field spectrometers (IFS) and spatial replication spectrometers, and their applications in fields such as astronomy, biology, and gas diffusion monitoring. It discusses the advantages and trade-offs of different IFS designs, such as slicer mirrors, lenslet arrays, and optical fiber bundles, and the challenges they face in terms of resolution, size, and complexity. For spatial replication SSI, the paper examines multi-channel beam-splitting and multi-aperture divided systems, detailing their optical components and spectral separation techniques. It also explores coded aperture SSI (CASSI), including double-dispersion, single-dispersion, and spatial-spectral coding methods, and their applications in spectral imaging. The paper further discusses pixelated filter array SSI, which uses periodic arrangements of pixel-level units to transmit or reflect specific spectra, and diffraction modulation SSI, which employs diffractive optical elements (DOEs) to introduce phase changes and achieve wavelength-dependent point spread functions. It highlights the advantages of diffractive lenses and diffractive optical networks in spectral imaging. Finally, the paper addresses combined modulation techniques, such as amplitude and wavelength modulation, phase and wavelength modulation, and amplitude and phase modulation, and their impact on system flexibility and performance. It concludes with an overview of spectral reconstruction technologies, including optimization algorithms and deep learning methods, and their applications in SSI.The paper provides a comprehensive overview of snapshot spectral imaging (SSI) technology, focusing on its system architecture, mathematical modeling, and recent advancements. SSI enables the capture of complete spectral information of objects in a very short time, making it suitable for dynamic observations in various fields such as environmental monitoring, medical diagnostics, and industrial inspection. The review highlights the evolution from early spatial-spectral mapping methods to more advanced techniques that encode and reconstruct various dimensions of light, including amplitude, phase, and wavelength. Metasurfaces, which offer unprecedented control over optical properties at sub-wavelength scales, have become a key component in SSI, reducing system size and improving efficiency. The paper discusses the integration of metasurfaces in SSI, detailing their optical principles, mathematical models, and modulation devices. It also explores different data acquisition schemes and detector capture modes, categorizing techniques into non-computational spatial-spectral mapping and computationally required coded reconstruction methods. The review covers various types of SSI systems, including integral field spectrometers (IFS) and spatial replication spectrometers, and their applications in fields such as astronomy, biology, and gas diffusion monitoring. It discusses the advantages and trade-offs of different IFS designs, such as slicer mirrors, lenslet arrays, and optical fiber bundles, and the challenges they face in terms of resolution, size, and complexity. For spatial replication SSI, the paper examines multi-channel beam-splitting and multi-aperture divided systems, detailing their optical components and spectral separation techniques. It also explores coded aperture SSI (CASSI), including double-dispersion, single-dispersion, and spatial-spectral coding methods, and their applications in spectral imaging. The paper further discusses pixelated filter array SSI, which uses periodic arrangements of pixel-level units to transmit or reflect specific spectra, and diffraction modulation SSI, which employs diffractive optical elements (DOEs) to introduce phase changes and achieve wavelength-dependent point spread functions. It highlights the advantages of diffractive lenses and diffractive optical networks in spectral imaging. Finally, the paper addresses combined modulation techniques, such as amplitude and wavelength modulation, phase and wavelength modulation, and amplitude and phase modulation, and their impact on system flexibility and performance. It concludes with an overview of spectral reconstruction technologies, including optimization algorithms and deep learning methods, and their applications in SSI.