Transitioning to energy-saving and renewable energy sources is impossible without accelerated development of hydrogen energy and hydrogen technologies. This review summarizes the state-of-the-art and recent advances of various hydrogen production processes, including but not limited to thermochemical and electrolytic processes. Their opportunities and limitations, operating conditions, and catalysts are discussed. Nowadays, most hydrogen is still produced by steam reforming of methane, its partial oxidation, or coal gasification. Considerable attention is also paid to natural gas pyrolysis. However, hydrogen produced using these technologies has a lot of impurities and needs additional purification. A series of technologies for hydrogen purification, including its filtration through palladium alloy membranes, and membrane catalysis, allowing hydrogen production and purification in one stage, are discussed. The main way to produce carbon-free hydrogen is water electrolysis using low-cost energy from nuclear or renewable sources. Both conventional and novel methods of hydrogen storage and transportation, which are an important part of the hydrogen economy, are reviewed. Biohydrogen production technologies are also discussed. Finally, prospects for further work in this field are provided. This review will be useful to researchers and manufacturers working in this field.