The linear least-squares prediction approach is applied to some problems in two-stage sampling from finite populations. A theorem giving the optimal (BLU) estimator and its error-variance under a general linear “superpopulation” model for a finite population is stated. This theorem is then applied to a model describing many populations whose elements are grouped naturally in clusters. Next, the probability model is used to analyze various conventional estimators and certain estimators suggested by the theory as alternatives to the conventional ones. Problems of design are considered, as are some consequences of regression-model failure.