Uncertainty Quantification (UQ) plays a pivotal role in computational modeling, enabling robust predictions and informed decision-making in the presence of uncertain input data. However, the high computational cost associated with repeated evaluations of complex models poses a significant challenge. Surrogate models offer a powerful solution to this issue, providing efficient and reliable approximations of full-order models. This minisymposium welcomes contributions on a broad spectrum of surrogate approaches with good error control and convergence guarantees and their use in UQ, including polynomial-based methods (e.g., stochastic Galerkin and collocation), rational approximations, Gaussian process regression and kernel methods, reduced basis techniques and dynamical low-rank approximations. Emphasis is placed to multilevel and multifidelity surrogate methods that leverage hierarchies of models or resolutions to further reduce computational effort while maintaining predictive accuracy. Applications are drawn primarily from computational mechanics (fluids and solids), quantum mechanics, and materials science.