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Chemical bonding and electronic structure of MgB2, a boron-based newly discovered superconductor, is studied using self-consistent band-structure techniques. Analysis of the transformation of the band structure for the hypothetical series of graphite–primitive graphite–primitive graphitelike boron–intercalated boron, shows that the band structure of MgB2 is graphitelike, with π bands falling deeper than in ordinary graphite. These bands possess a typically delocalized and metallic, as opposed to covalent, character. The in-plane σ bands retain their two-dimensional (2D) covalent character, but exhibit a metallic hole-type conductivity. The coexistence of 2D covalent in-plane and three-dimensional (3D) metallic-type interlayer conducting bands is a peculiar feature of MgB2. We analyze the 2D and 3D features of the band structure of MgB2 and related compounds, and their contributions to conductivity.