Bifunctional catalysts for oxygen reduction reaction (ORR) and 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 such bifunctionality to be enginnered within one catalyst, together with improved cost-effectiveness and durablility. Herein, by closely coupled computational design and experimental development, highly effective bifunctionality is 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 the 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 such sites by one-pot synthesis to tailor the PNGF. The resulting catalyst reaches 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, respectively. Its combined ORR and OER overpotential of 705 mV is significantly lower than those reported previously for metal-free bifunctional catalysts.