Primary and rechargeable Zn-air batteries could be ideal energy storage devices with high energy and power density, high safety and economic viability. Active and durable electrocatalysts on the cathode side are required to catalyse oxygen reduction reaction during discharge and oxygen evolution reaction during charge for rechargeable batteries. Here we developed advanced primary and rechargeable Zn-air batteries with novel CoO/carbon nanotube hybrid oxygen reduction catalyst and Ni-Fe-layered double hydroxide oxygen evolution catalyst for the cathode. These catalysts exhibited higher catalytic activity and durability in concentrated alkaline electrolytes than precious metal Pt and Ir catalysts. The resulting primary Zn-air battery showed high discharge peak power density ~265 mW cm-2, current density ~200 mA cm-2 at 1 V and energy density >700 Wh kg-1. Rechargeable Zn-air batteries in a tri-electrode configuration exhibited an unprecedented small charge-discharge voltage polarization of ~0.70 V at 20 mA cm-2, high reversibility and stability over long charge and discharge cycles.