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Currently, identifying suitable oxygen reduction reaction (ORR) catalysts in novel two-dimensional (2D) materials has attracted more and more research attention. Here, we have studied the catalytic activities of 2D h-BeN2 and MgN2 monolayers for ORR by using first-principles calculations. The calculated results reveal that the direct quasiparticle bandgap of BeN2 monolayer is 3.32 eV, and the indirect bandgap of MgN2 is 3.42 eV. 2D h-BeN2 and MgN2 exhibit high exciton binding energies of 1.07 and 0.83 eV respectively, and their optical properties are determined by bound exciton transitions due to the strong quantum confinement effects. Importantly, h-BeN2 and MgN2 monolayers with positive-charged (+1.6 e) metal atom (Be/Mg) on the surface exhibit excellent adsorption ability for O2 and ORR intermediates, and show better CO tolerance than Pt(111). The calculated free energy plots are always downhill for ORR catalyzed by BeN2 in both acid and alkaline environments, and by MgN2 in alkaline environments. The detailed reaction mechanism analyses show that high-efficient four-electron pathway is the optimal pathway for ORR catalyzed by BeN2 in acid environments. Surprisingly, there is a low overpotential of 0.45 eV for ORR catalyzed by BeN2 in the acid solution and no overpotential in the alkaline solution. Our studies found for the first time that 2D h-BeN2 shows huge potential as a non-precious metal ORR catalyst in acid and alkaline environments.