The paper introduces Gaussian predictive process models for large spatial data sets. These models are designed to address computational challenges in fitting hierarchical spatial models, which often involve expensive matrix operations that become infeasible for large data sets. The key idea is to use predictive process models that project the spatial process onto a lower-dimensional subspace, thereby reducing computational burden. This approach allows for flexibility in accommodating non-stationary, non-Gaussian, multivariate, and spatiotemporal processes. The models are based on the concept of a predictive process derived from a parent spatial process, where the predictive process is a linear transformation of the parent process's realizations at a set of specified locations (knots). This method enables efficient computation by reducing the dimensionality of the problem, making it feasible for large data sets. The paper discusses the theoretical properties of these predictive processes, provides a computational template for their implementation, and illustrates the approach with simulated and real data examples. It also addresses the extension of these models to multivariate and spatiotemporal settings, and discusses Bayesian implementation and computational issues, including the use of Gibbs sampling and Metropolis steps for parameter estimation. The models are shown to be effective in handling large spatial data sets with complex spatial dependencies.The paper introduces Gaussian predictive process models for large spatial data sets. These models are designed to address computational challenges in fitting hierarchical spatial models, which often involve expensive matrix operations that become infeasible for large data sets. The key idea is to use predictive process models that project the spatial process onto a lower-dimensional subspace, thereby reducing computational burden. This approach allows for flexibility in accommodating non-stationary, non-Gaussian, multivariate, and spatiotemporal processes. The models are based on the concept of a predictive process derived from a parent spatial process, where the predictive process is a linear transformation of the parent process's realizations at a set of specified locations (knots). This method enables efficient computation by reducing the dimensionality of the problem, making it feasible for large data sets. The paper discusses the theoretical properties of these predictive processes, provides a computational template for their implementation, and illustrates the approach with simulated and real data examples. It also addresses the extension of these models to multivariate and spatiotemporal settings, and discusses Bayesian implementation and computational issues, including the use of Gibbs sampling and Metropolis steps for parameter estimation. The models are shown to be effective in handling large spatial data sets with complex spatial dependencies.