This dissertation, comprising two distinct papers, investigates the prediction and sampling of spatial processes, where the data are contaminated with measurement error;In the first paper, we show that a geostatistical model can provide a powerful way of predicting unknown parts of some spatial phenomenon. The prediction problem is multivariate in the sense that one wishes to predict at multiple spatial locations. The results presented in this paper offer compelling evidence that a geostatistical model should be incorporated into spatial sampling and analysis, where possible. Even when the observable process is contaminated with measurement error, there is a straightforward way to filter it out by appropriately modifying the spatial prediction equations. Our results show that a geostatistical analysis of a certain class of non-clustered designs, whether simple-random, stratified-random, or systematic-with-a-random-start, performs extremely well with respect to design-based optimality criteria. In contrast, clustered designs, corresponding to repeated sampling from "representative sites", perform very poorly. One important aspect of our study is the prediction of spatial statistics defined over small areas (called local regions), that are subsets of a global region over which a network of sampling sites is chosen. Under circumstances where both local and nonlinear functions of the process are to be predicted, it is demonstrated that appropriate geostatistical analyses perform very well, irrespective of the (non-clustered) sampling design;In the second paper, a spatial model that explicitly includes a measurement-error component is proposed, to accommodate the fact that data is almost always contaminated with measurement error. Linear predictors can easily accommodate this measurement-error component, but this is not true of nonlinear predictors, which may be substantially biased if the measurement-error variance is large. For the prediction of nonlinear functionals of spatial processes, constrained kriging is examined in detail, especially with regard to its existence conditions, its geometric interpretation, its applicability to certain "nonspatial" problems, and its relationship with conditional simulation. The theory supporting constrained kriging is extended to the "covariance-matching" case where multiple predictions are required simultaneously.