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As a branch of machine learning, Gaussian process regression (GPR) has received increasing attention in recent years and is widely used in many fields. GPR is used for modeling nonlinear systems and can automatically trade-off between model complexity and accuracy. However, due to its high computational complexity, it is difficult to be directly applied to learning tasks with large data sizes. Therefore, many approximation methods are developed to reduce its computational cost. According to whether the training data is divided into subsets, the approximation methods of GPR can be categorized as global and local approximations. This article first describes the theoretical basis of GPR, analyzes these two approximation methods; Then its applications in practice are introduced, especially in the fields of soft sensing and control; Finally, a summary and a prospect of its future research direction are given.