Heretofore disconnected experimental observations are combined with a theoretical study to develop a model of the chemical composition of the edges of graphene sheets in both flat and curved sp(2)-hybridized carbon materials. It is proposed that under ambient conditions a significant fraction of the oxygen-free edge sites are neither H-terminated nor unadulterated sigma free radicals, as universally assumed. The zigzag sites are carbene-like, with the triplet ground state being most common. The armchair sites are carbyne-like, with the singlet ground state being most common. This proposal is not only consistent with the key electronic properties and surface (re)activity behavior of carbons, but it can also explain the recently documented and heretofore puzzling ferromagnetic properties of some impurity-free carbon materials.