Based upon the similarities in properties between carbon- and BN-based (BN=boron nitride) materials, we propose that BN-based nanotubes can be stable and study their electronic structure. A simple Slater-Koster tight-binding scheme has been applied. All the BN nanotubes are found to be semiconducting materials. The band gaps are larger than 2 eV for most tubes. Depending on the helicity, the calculated band gap can be direct at [Gamma] or indirect. In general, the larger the diameter of the nanotube the larger the band gap, with a saturation value corresponding to the calculated local-density-approximation band gap of hexagonal BN. The higher ionicity of BN is important in explaining the electronic differences between these tubes and similar carbon nanotubes.