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. In addition, as described in the final chapter, fixed-domain asymptotics provides the potential basis for a theory of kriging with unknown dependence structures that is both mathematically sound and relevant to the practice of kriging.