Bifunctional catalysts for the oxygen reduction reaction (ORR) and the oxygen evolution reaction (OER) are highly desirable for rechargeable metal–air batteries and regenerative fuel cells. However, the commercial oxygen electrocatalysts (mainly noble metal based) can only exhibit either ORR or OER activity and also suffer from inherent cost and stability issues. It remains challenging to achieve efficient ORR and OER bifunctionality on a single catalyst. Metal-free structures offer relatively large scope for this bifunctionality to be engineered within one catalyst, together with improved cost-effectiveness and durability. Herein, by closely coupled computational design and experimental development, highly effective bifunctionality was achieved in a phosphorus and nitrogen co-doped graphene framework (PNGF) – with both ORR and OER activities reaching the theoretical limits of metal-free catalysts, superior to their noble metal counterparts in both (bi)functionality and durability. In particular, with the identification of active P–N sites for OER and N-doped sites for ORR, we successfully intensified these sites by one-pot synthesis to tailor the PNGF. The resulting catalyst achieved an ORR potential of 0.845 V vs. RHE at 3 mA cm−2 and an OER potential of 1.55 V vs. RHE at 10 mA cm−2. Its combined ORR and OER overpotential of 705 mV is much lower than those previously reported for metal-free bifunctional catalysts.