Kriging, which is the geostatistical term for optimal linear prediction of spatial processes, is widely used in geology, hydrology, environmental monitoring, and other fields to interpolate spatial data. Despite its widespread usage, there is as yet no rigorous theoretical basis for the performance of kriging when some aspect of the dependence structure of the spatial process must be estimated, which is generally the case in practice.<BR>Synthesizing past work of the author with many new results, this monograph proposes using fixed-domain asymptotics, in which one considers an increasing number of observations in a fixed and bounded observation domain, as the best way to study kriging. This approach yields an understanding of the critical relationship between the properties of kriging predictors and the local behavior of the spatial process.