In the face of the electrification trend in transportation, the internal combustion engine (ICE) is expected to continue playing a vital role in generating electricity for power systems or directly propelling vehicles in certain sectors. However, ICEs are also under significant pressure to achieve carbon neutrality, with the key lying in carbon-free fuels. Ammonia, compared to hydrogen, offers advantages in terms of hydrogen-carrying capacity, storage and transportation convenience, and safety, making it a promising carbon-free fuel for large-scale use in ICEs. Nonetheless, ammonia's combustion inertness poses challenges for its application, requiring efforts to enhance its combustion. Hydrogen, as a carbon-free and highly reactive fuel, serves as a powerful combustion promoter, maximizing the carbon-free effect of ammonia. Furthermore, on-board ammonia decomposition can produce hydrogen, ensuring a stable hydrogen supply and enabling ammonia-hydrogen synergy combustion while carrying only ammonia. This ammonia-hydrogen synergy combustion, based on on-board hydrogen production, presents a highly promising development direction for ammonia engines. When combined with hybridization, it further enhances the overall energy efficiency of ammonia. The objective of this paper is to review recent advancements in ammonia-hydrogen engines, covering topics such as ignition methods and combustion strategies, fuel supply, pollutants, and after-treatment. Based on this review, a conceptual ammonia-hydrogen engine for hybrid power systems is proposed. This engine ignites the ammonia-hydrogen mixture in the main chamber using hydrogen active jet ignition, achieving spark-assisted compression ignition. Technical measures for efficient engine combustion, synergistic utilization of exhaust heat for hydrogen production, and effective after-treatment of NOx, unburned NH3, and N2O are discussed. At last, some perspectives on the development of ammonia-hydrogen engines are also presented.