It is known that bulk ZnO is a nonmagnetic material. However, the electronic band structure of ZnO is severely distorted when the ZnO is in the shape of a very thin plate with its dimension along the c-axis reduced to a few nanometers while keeping the bulk scale sizes in the other two dimensions. We found that the chemically synthesized ZnO nanoplates exhibit magnetism even at room temperature. First-principles calculations show a growing asymmetry in the spin distribution within the distorted bands formed from Zn (3d) and O (2p) orbitals with the reduction of thickness of the ZnO nanoplates, which is suggested to be responsible for the observed magnetism. In contrast, reducing the dimension along the a-or b-axes of a ZnO crystal does not yield any magnetism for ZnO nanowires that grow along c-axis, suggesting that the internal electric field produced by the large {0001} polar surfaces of the nanoplates may be responsible for the distorted electronic band structures of thin ZnO nanoplates.