In recent years, hydrogen-rich compounds under extremely high pressure have become the hot target materials for high-temperature superconductors. At present, two landmark progresses have been made in this field. Covalent H₃S hydrogen-rich superconductors (T_c = 200 K) and ionic hydrogen-rich superconductors with hydrogen-cage structure, such as LaH₁₀ (T_c = 260 K, –13 ℃), YH₆ and YH₉, have been successively synthesized, setting a new record of superconducting temperature. These studies have given rise to the hope of discovering room-temperature superconductors in hydrogen-rich compounds under high pressure. This paper focuses on the progress of hydrogen-rich superconductors with high critical temperature under high pressure, discusses the physical mechanism of high-temperature superconductivity in hydrogen-rich compounds, provide an outlook on the possibility of discovering room-temperature superconductors in hydrogen-rich compounds in the future, and offer the candidate system for high superconductivity in multiple hydrogen-rich compounds.