The paper introduces Gaussian predictive process models for large spatial data sets, addressing the computational challenges associated with fitting hierarchical spatial models. The authors propose a method that projects the original spatial process onto a lower-dimensional subspace, reducing the computational burden while maintaining flexibility in modeling non-stationary, non-Gaussian, multivariate, and spatiotemporal processes. The approach is theoretically grounded and practical, with a focus on computational efficiency. The paper discusses the theoretical aspects, provides a computational template, and illustrates the method with simulated and real data sets. The authors also explore extensions to multivariate processes and spatiotemporal contexts, demonstrating the effectiveness of the predictive process model in handling complex spatial data.The paper introduces Gaussian predictive process models for large spatial data sets, addressing the computational challenges associated with fitting hierarchical spatial models. The authors propose a method that projects the original spatial process onto a lower-dimensional subspace, reducing the computational burden while maintaining flexibility in modeling non-stationary, non-Gaussian, multivariate, and spatiotemporal processes. The approach is theoretically grounded and practical, with a focus on computational efficiency. The paper discusses the theoretical aspects, provides a computational template, and illustrates the method with simulated and real data sets. The authors also explore extensions to multivariate processes and spatiotemporal contexts, demonstrating the effectiveness of the predictive process model in handling complex spatial data.