The catalyst in the oxygen electrode is the core component of the aqueous metal-air battery, which plays a vital role in the determination of the open circuit potential, energy density, and cycle life of the battery. For rechargeable aqueous metal-air batteries, the catalyst should have both good oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) catalytic performance. Compared with precious metal catalysts, non-precious metal materials have more advantages in terms of abundant resource reserves and low prices. Over the past few years, great efforts have been made in the development of non-precious metal bifunctional catalysts. This review selectively evaluates the advantages, disadvantages and development status of recent advanced materials including pure carbon materials, carbon-based metal materials and carbon-free materials as bifunctional oxygen catalysts. Preliminary improvement strategies are formulated to make up for the deficiency of each material. The development prospects and challenges facing bifunctional catalysts in the future are also discussed.