It remains a grand challenge to achieve both high activity and durability in Pt electrocatalysts for oxygen reduction reaction (ORR) in fuel cells. Here we develop a class of Pt highly concave cubic (HCC) nanocrystals, which are enriched with high-index facets, to enable high ORR activity. The durability of HCC nanocrystals can be significantly improved via assembly with graphene. Meanwhile, the unique hybrid structure displays further enhanced specific activity, which is 7-fold greater than the state-of-the-art Pt/C catalysts. Strikingly, it exhibits impressive performance in terms of half-wave potential (E1/2). The E1/2 of 0.967 V at the Pt loading as low as 46 μg cm(-2), which stands as 63 mV higher than that of the Pt/C catalysts, is slightly superior to the record observed for the most active porous Pt-Ni catalyst in literature. This work paves the way to designing high-performance electrocatalysts by modulating their surface and interface with loading substrates.