The first-principles calculations have been performed to understand the origin of magnetism in undoped GaN thin films. The results show that Ga vacancy, rather than that of N contributes the observed magnetism, and the magnetic moments mainly come from the unpaired 2p electrons at nearest-neighbor N atoms of the Ga vacancy. Calculations and discussions are also extended to bare and passivated GaN nanowires, We find that per Ga vacancy on the surface sites products the total magnetic moment of 1.0 mu(B) while that inside of the nanowires can lead to the formation of a net moment of 3.0 mu(B). The coupling between two Ga vacancies is also studied and we found that the coupling is ferromagnetic coupling. The surface passivation with hydrogen is shown to strongly enhance the ferromagnetism. Our theoretical study not only demonstrates that GaN nanowire can be magnetic even without transitionmetal doping, but also suggests that introducing Ga vacancy is a natural and an effective way to fabricate low-dimensional magnetic GaN nanostructures. (C) 2010 Elsevier B.V. All rights reserved.