In this study, the electronic properties of ZnO nanoribbons with zigzag edges (ZZnONr) have been investigated with Density Functional Theory (DFT). After a geometric optimization, the electronic band structures, the density of states (DOS) of ZZnONr passivated with Hydrogen (H) and Magnesium (Mg) atoms were calculated ZZnONr. It is shown that the increasing width of ZZnONrs has led to a decrement in energy band gap of the studied structures. While ZZnONr passivated with Mg for Zn-rich edge have not been shown a spin dependency, the structure passivated with Mg for O-rich edge have exhibited spin-dependent band structure. The energetically most stable structures have been determined as ZZnONr passivated with Mg for Zn-rich edge. ZZnONr passivated with Mg atoms for both edges have a graphene-like band structure especially for 8 and 10 atom width structures and this property of ZZnONrs could be important in terms of the electron transport for ZZnONrs.