The electronic structures of the intrinsic and nitrogen-doped silicon carbide nanotubes （SiCNTs） have been calculated by first-principles approach based on the density functional theory. The intrinsic （8, 0） SiCNT is a direct band-gap semiconductor with a gap value of 0.94 eV. The band-gap of the SiCNT with the doping concentration of nitrogen being 1.56% and 3.12％ is narrowed to 0.83 eV and 0.74 eV， respectively. The narrowing of the band-gap is the result of the weakening of the Si-N bonds compared with the corresponding Si-C bonds, which can be seen by comparing the charge density difference of the intrinsic SiCNT with that of the nitrogen-doped nanotube.