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Hyperspectral imaging is an important technique in remote sensing which is characterized by high spectral resolutions. With the advent of new hyperspectral remote sensing missions and their increased temporal resolutions, the availability and dimensionality of hyperspectral data is continuously increasing. This demands fast processing solutions that can be used to compress and/or interpret hyperspectral data onboard spacecraft imaging platforms in order to reduce downlink connection requirements and perform a more efficient exploitation of hyperspectral data sets in various applications. Over the last few years, reconfigurable hardware solutions such as field-programmable gate arrays (FPGAs) have been consolidated as the standard choice for onboard remote sensing processing due to their smaller size, weight, and power consumption when compared with other high-performance computing systems, as well as to the availability of more FPGAs with increased tolerance to ionizing radiation in space. Although there have been many literature sources on the use of FPGAs in remote sensing in general and in hyperspectral remote sensing in particular, there is no specific reference discussing the state-of-the-art and future trends of applying this flexible and dynamic technology to such missions. In this work, a necessary first step in this direction is taken by providing an extensive review and discussion of the (current and future) capabilities of reconfigurable hardware and FPGAs in the context of hyperspectral remote sensing missions. The review covers both technological aspects of FPGA hardware and implementation issues, providing two specific case studies in which FPGAs are successfully used to improve the compression and interpretation (through spectral unmixing concepts) of remotely sensed hyperspectral data. Based on the two considered case studies, we also highlight the major challenges to be addressed in the near future in this emerging and fast growing research area.