Organometallic complexes with metal-nitrogen/carbon (M-N/C) coordination are the most important alternatives to precious metal catalysts for oxygen reduction and evolution reactions (ORR and OER) in energy conversion devices. Here, we designed and developed a range of molecule-level graphitic carbon nitride (g-C3N4) coordinated transition metals (M-C3N4) as a new generation of M-N/C catalysts for these oxygen electrode reactions. As a proof-of-concept example, we conducted theoretical evaluation and experimental validation on a cobalt-C3N4 catalyst with a desired molecular configuration, which possesses comparable electrocatalytic activity to that of precious metal benchmarks for the ORR and OER in alkaline media. The correlation of experimental and computational results confirms that this high activity originates from the precise M-N2 coordination in the g-C3N4 matrix. Moreover, the reversible ORR/OER activity trend for a wide variety of M-C3N4 complexes has been constructed to provide guidance for the molecular design of this promising class of catalysts.