A resurgence of interest in aqueous acid-based redox flow batteries with anion exchange membranes has been revitalized by the promising progress in this type of membrane. The striking feature of this type of flow battery is that the use of anion exchange membrane enables a significant reduction in crossover of electroactive species (e.g., vanadium ions), thereby improving the coulombic efficiency. This merit is derived from the Donnan exclusion effect between positively charged functional groups grafted on the membranes and electroactive species, which are normally positive ions. This article presents a comprehensive review of the development of anion exchange membranes with different categories of functional groups for aqueous acid-based redox flow batteries. In the meantime, ion-solvating polymers and amphoteric ion exchange membranes that share the same mechanism to reduce crossover of electroactive species and their use in aqueous acid-based redox flow batteries are also reviewed. Subsequently, the critical issues preventing the widespread use of anion exchange membranes, including poor chemical stability and uncertain conduction mechanism, are elaborated. The perspective for the further development of anion exchange membranes is summarized in the final section. We expect to shed light on the development direction of anion exchange membranes for aqueous acid-based redox flow batteries.