Abstract This article presents an overview of the fundamental optical phenomena behind compressive spectral imaging systems based on coded apertures. The key mathematical concepts embodying the sensing and reconstruction methods, the framework developed to design optimal coded apertures together with the computational spectral imaging strategies for classification, unmixing, and spectral selectivity purposes are also presented. Special attention is given for describing and discussing many practical aspects of compressive spectral imaging, including discretization models of the optical system, optimal parameters design, and physical limitations. The performance of the sensing strategies and computational methods is illustrated in this article with real data and imagery for different applications.