Various properties of ceramics can be significantly influenced by the presence of grain boundaries. The influence on the properties is closely related to the grain-boundary atomic structures. As different grain boundaries have different atomic structure, different grain boundaries have different influence on the properties. It is difficult to examine the atomic structure and properties of individual grain boundaries in ceramics. In order to understand the atomic-structure-property relationships, well-defined single grain boundaries should be characterized. In the present paper, we review our recent results on the investigations of atomic structures and electrical properties of ZnO single grain boundaries. The relationships between the atomic structures and the electrical properties were investigated using ZnO bicrystals, whose grain-boundary orientation relationship and grain-boundary planes can be arbitrarily controlled. The discussion focuses on the microscopic origin of nonlinear current-voltage (I-V) characteristics across ZnO grain boundaries. High-resolution transmission electron microscopy (HRTEM) observations and lattice-statics calculations revealed the atomic structures of the undoped ZnO [0001] Sigma 7 and Sigma 49 grain boundaries, enabling a comparison between coincidence site lattice (CSL) boundaries with small and large periodicity. These grain boundaries contained the common structural units (SUs) featuring atoms with coordination numbers that are unusual in ZnO. The Sigma 49 boundary was found to have characteristic arrangement of the SUs, where two kinds of the SUs are alternatively formed. It is considered that the characteristic arrangement was formed to effectively relax the local strain in the vicinity of the boundary. Such a relaxation of local strain is considered to be one of dominant factors to determine the SU arrangements along grain boundaries. I-V measurements of the undoped ZnO bicrystals showed linear I-V characteristics. Although the coordination and bond lengths of atoms in the grain boundaries differ from those in the bulk crystal, this does apparently not generate deep unoccupied states in the band gap. This indicates that atomic structures of undoped ZnO grain boundaries are not responsible for the nonlinear I-V characteristics of ZnO ceramics. On the other hand, the nonlinear I-V characteristic appeared when doping the boundaries with Pr. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image of Pr-doped boundaries revealed that Pr segregates to specific atomic columns, substituting Zn at the boundary. However, the Pr itself was not the direct origin of the nonlinear I-V characteristics, as the Pr existed in the three-plus state and would not produce acceptor states in the boundary. First-principles calculations revealed that Pr doping instead promotes the formations of acceptor-like native defects, such as Zn vacancies. We believe that such acceptor-like native defects are microscopic origin of the nonlinear I-V characteristics. Investigations of various types of grain boundaries in the Pr and Co-codoped ZnO bicrystals indicated that the amounts of Pr segregation and the nonlinear I-V characteristics significantly depend on the grain-boundary orientation relationship. Larger amount of Pr segregation and, as a result, higher nonlinearity in I-V characteristics was obtained for incoherent boundaries. This indicates that Pr doping to incoherent boundaries is one of the guidelines to design the single grain boundaries with highly nonlinear I-V characteristics Finally, a Pr and Co-codoped bicrystal with an incoherent boundary was fabricated to demonstrate a highly nonlinear I-V characteristic. This result indicates that ZnO single-grain-boundary varistors can be designed by controlling grain-boundary atomic structures and fabrication processes. Summarizing, our work firstly enabled us to gain a deeper understanding for the atomic structure of ZnO grain boundaries. Secondly, we obtained important insight into the origin of nonlinear I-V characteristics across the ZnO grain boundaries. And, finally, based on these results, we demonstrated the potential of this knowledge for designing and fabricating ZnO single-grain-boundary varistors.