Hydrogen sensors, especially those operating at high temperatures, are essential tools for the emerging hydrogen economy. Monitoring hydrogen under process conditions to control the reactions for detecting confined species is crucial to the safe, widespread use and public acceptance of hydrogen as fuel. Hydrogen sensors must have a sensitivity ranging from traces of hydrogen (parts per million (ppm)) up to levels near the lower explosive limit (LEL = 4% H2 in the air) for safety reasons. Furthermore, they need to operate in cryogenic, ambient, and high-temperature environments. Herein, emphasis is given to hydrogen sensors based on solid oxide electrolytes (operating at high temperatures), in particular oxygen ion and proton conductors. The review is devoted to potentiometric, amperometric, and combined amperometric-potentiometric hydrogen sensors. Experimental results already reported in the international literature are presented and analyzed to reveal the configuration, principle of operation, and the applied solid electrolytes and electrodes of the high-temperature hydrogen sensors. Additionally, an amperometric sensor able to detect hydrogen and steam in atmospheric air through a two-stage procedure is presented and thoroughly discussed. The discussion reveals that high-temperature hydrogen sensors face different challenges in terms of the electrodes and solid electrolytes to be used, depending on the operating principle of each sensor type.