Abstract: Solid oxide fuel cells (SOFCs) are regarded to be a key clean energy system to convert chemical energy (e.g. H₂ and O₂) into electrical energy with high efficiency, low carbon footprint, and fuel flexibility. The electrolyte, typically doped zirconia, is the "state of the heart" of the fuel cell technologies, determining the performance and the operating temperature of the overall cells. Yttria stabilized zirconia (YSZ) have been widely used in SOFC due to its excellent oxide ion conductivity at high temperature. The composition and temperature dependence of the conductivity has been hotly studied in experiment and, more recently, by theoretical simulations. The characterization of the atomic structure for the mixed oxide system with different compositions is the key for elucidating the conductivity behavior, which, however, is of great challenge to both experiment and theory. This review presents recent theoretical progress on the structure and conductivity of YSZ electrolyte. We compare different theoretical methods and their results, outlining the merits and deficiencies of the methods. We highlight the recent results achieved by using stochastic surface walking global optimization with global neural network potential (SSW-NN) method, which appear to agree with available experimental data. The advent of machine-learning atomic simulation provides an affordable, efficient and accurate way to understand the complex material phenomena as encountered in solid electrolyte. The future research directions for design better electrolytes are also discussed.