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The data-driven material models have attracted many researchers recently, as they could directly use material data. However, there are limited studies about material uncertainty in previous data-driven models. This thesis proposes a new Gaussian Process Regression (GPR)-based approach to capture the material behaviour and the associated material uncertainty from the dataset. The GPR approach is firstly used for the nonlinear elastic behaviour. The obtained GPR-based model is verified by the material datasets. Then, an improved GPR model, called the Heteroscedastic Sparse Gaussian Process Regression (HSGPR) model, is applied for the plastic flow behaviour. The flow stress predicted by the HSGPR model also agrees with the experiments. As a new data-driven material model is introduced, the associated frameworks, which implement the GPR-based model and HSGPR-based model into the finite element method for structural reliability analysis, are developed. The frame problem is used to demonstrate the GPR-based model in the elastic stochastic structural analysis, while the beam and punch problems validate the HSGPR-based model in the plastic stochastic structural analysis. It is concluded that the GPR-based approach can accurately identify both the elastic and plastic stochastic structural responses. To consider the possible correlation of the stochastic material behaviours, a novel GPR-based approach, which combines the HSGPR model with the Proper Orthogonal Decomposition (POD) algorithm, is proposed. Two case studies on the metal strength and the rock joint behaviour have demonstrated that the material behaviours correlation can be effectively retained in the POD-HSGPR-based model. As indicated by its application in a rock slope problem, it is critical to consider the material properties correlation for the accurate evaluation of structural reliability.