Modern hyperspectral imaging systems produce huge datasets potentially conveying a great abundance of information; such a resource, however, poses many challenges in the analysis and interpretation of these data. Deep learning approaches certainly offer a great variety of opportunities for solving classical imaging tasks and also for approaching new stimulating problems in the spatial–spectral domain. This is fundamental in the driving sector of Remote Sensing where hyperspectral technology was born and has mostly developed, but it is perhaps even more true in the multitude of current and evolving application sectors that involve these imaging technologies. The present review develops on two fronts: on the one hand, it is aimed at domain professionals who want to have an updated overview on how hyperspectral acquisition techniques can combine with deep learning architectures to solve specific tasks in different application fields. On the other hand, we want to target the machine learning and computer vision experts by giving them a picture of how deep learning technologies are applied to hyperspectral data from a multidisciplinary perspective. The presence of these two viewpoints and the inclusion of application fields other than Remote Sensing are the original contributions of this review, which also highlights some potentialities and critical issues related to the observed development trends.