A systematic review has been made of the geometric and electronic properties of graphene-related systems. The first-principles calculations are used to explore the rich essential properties. Both structure-enriched and adatom-doped graphenes can exhibit the semiconducting, semimetallic, and metallic behaviors. The dramatic transition of distinct band structures occurs during the transformation of stacking configuration or the change of geometric curvature. Two pairs of vertical Dirac cones, parabolic bands, and non-vertical Dirac cones are transformed into each other. For graphene oxides (GOs), the competition or cooperation among the critical chemical bondings in C-C, O-O, and C-O bonds leads to the diversified properties. The distinct orbital hybridizations are identified from the atom-related energy bands, the spatial charge density, and the orbital-projected density of states (DOS). Halogenated graphenes, with the top-side adatom adsorption, exhibit the diverse and unique chemical bondings, especially for the great differences between fluorination and other halogenations.