Density-functional calculations are used to predict the stability and electronic structures of GaN nanotubes. Strain energies of GaN nanotubes are comparable to those of carbon nanotubes, suggesting the possibility for the formation of GaN nanotubes. The zigzag nanotube is a semiconductor with direct band gap, whereas the armchair nanotube has an indirect band gap. The band gaps decrease with decreasing diameter, contrary to the case of carbon nanotubes. We further discuss possible ways of synthesizing GaN nanotubes in conjunction with carbon nanotubes.