Bifunctional catalysts can facilitate two different electrochemical reactions with conflicting characteristics. Here, a highly reversible bifunctional electrocatalyst for rechargeable zinc-air batteries (ZABs) is reported featuring a "core-shell structure" in which N-doped graphene sheets wrap around vanadium molybdenum oxynitride nanoparticles. Single Mo atoms are released from the particle core during synthesis and anchored to electronegative N-dopant species in the graphitic shell. The resultant Mo single-atom catalysts excel as active oxygen evolution reaction (OER) sites in pyrrolic-N and as active oxygen reduction reaction (ORR) sites in pyridinic-N environments. ZABs with such bifunctional and multicomponent single-atom catalysts deliver high power density (& AP;376.4 mW cm(-2)) and long cycle life of over 630 h, outperforming noble-metal-based benchmarks. Flexible ZABs that can tolerate a wide range of temperatures (-20 to 80 & DEG;C) under severe mechanical deformation are also demonstrated