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The article gives a comprehensive overview of hydrogen storage in carbon nanostructures, including experimental results and theoretical calculations. Soon after the discovery of carbon nanotubes in 1991, different research groups succeeded in filling carbon nanotubes with some elements, and, therefore, the question arose of filling carbon nanotubes with hydrogen by possibly using new effects such as nano-capillarity. Subsequently, very promising experiments claiming high hydrogen storage capacities in different carbon nanostructures initiated enormous research activity. Hydrogen storage capacities have been reported that exceed the benchmark for automotive application of 6.5 wt% set by the U.S. Department of Energy. However, the experimental data obtained with different methods for various carbon nanostructures show an extreme scatter. Classical calculations based on physisorption of hydrogen molecules could not explain the high storage capacities measured at ambient temperature, and, assuming chemisorption of hydrogen atoms, hydrogen release requires temperatures too high for technical applications. Up to now, only a few calculations and experiments indicate the possibility of an intermediate binding energy. Recently, serious doubt has arisen in relation to several key experiments, causing considerable controversy. Furthermore, high hydrogen storage capacities measured for carbon nanofibers did not survive cross-checking in different laboratories. Therefore, in light of today's knowledge, it is becoming less likely that at moderate pressures around room temperature carbon nanostructures can store the amount of hydrogen required for automotive applications.