We use first principles calculations to investigate the magnetic properties of zinc oxide nanoribbons with zigzag-terminated edges. The polarized spin density of states is calculated as a function of the nanoribbons width and thickness. All nanoribbons formed by a single layer exhibit a magnetic behavior independently of the width. By analyzing the charge density and spin density, we determine that the oxygen-dominated edge exhibits unpaired spins. When the thickness of the ribbons is increased, a magnetic moment is observed only for specific thicknesses.