Spatial statistics is concerned with the analysis of data that have spatial locations associated with them, and those locations are used to model statistical dependence between the data. The spatial data are treated as a single realization from a probability model that encodes the dependence through both fixed effects and random effects, where randomness is manifest in the underlying spatial process and in the noisy, incomplete measurement process. The focus of this review article is on the use of basis functions to provide an extremely flexible and computationally efficient way to model spatial processes that are possibly highly nonstationary. Several examples of basis-function models are provided to illustrate how they are used in Gaussian, non-Gaussian, multivariate, and spatio-Temporal settings, with applications in geophysics. Our aim is to emphasize the versatility of these spatial-statistical models and to demonstrate that they are now center-stage in a number of application domains. The review concludes with a discussion and illustration of software currently available to fit spatial-basis-function models and implement spatial-statistical prediction.