Since the discovery of 4.2 K superconductivity in mercury, the search for room-temperature superconductivity has been a hot topic in the field of condensed matter physics. In recent years, scientists have discovered a series of high-temperature superconductivity, represented by covalent H₃S (superconducting transition temperature T_c=203 K) and ionic LaH₁₀ (T_c=250 K) and CaH₆ (T_c=215 K) under high pressures. These works have successively broken superconducting temperature records, opening a new era in search for room-temperature superconductivity in hydrogen-rich compounds. This paper focuses on the progress of theoretical prediction, experimental synthesis, and characterization in binary and ternary hydrogen-rich superconductors with high critical temperature under high pressure. Furthermore, it addresses the challenges and potential avenues in the quest for room-temperature superconductors in hydrogen-rich compounds.