Bayesian surrogate models play a central role in the analysis of computationally expensive physics-based simulators, particularly when uncertainty quantification (UQ) is a primary objective. This review provides a high-level overview of commonly used Bayesian surrogate modeling methods for high-dimensional computer simulations, with emphasis on both methodological foundations and practical applications. We review the advantages of Bayesian surrogates, the challenges posed by high-dimensional outputs, and two broad classes of methods: Gaussian process models, including scalable approximations, and Bayesian basis function regression approaches such as polynomial chaos expansions, Bayesian multivariate adaptive regression splines, Bayesian projection pursuit regression, and Bayesian additive regression trees. We also review strategies for handling high-dimensional simulator outputs, of which dimension-reduction approaches based on low-dimensional output representations are most common. Finally, we highlight key applications of Bayesian surrogate modeling in forward UQ, global sensitivity analysis, and Bayesian model calibration, and demonstrate a practical workflow utilizing these methods in simulated and real data analyses. We conclude by identifying open challenges and future research directions related to method selection, computational scalability, predictive accuracy, and UQ fidelity.

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