Regression estimators for the finite population mean constructed under superpopulation models are considered. The conditions under which the model-based estimator is design-consistent are presented. Methods of augmenting the regression model to produce a design consistent estimator are discussed. It is shown that adding a specified column to the model-based regression estimator gives a design consistent regression estimator with the generalized least squares estimator for the vector of regression coefficients.;The regression estimator that is the best linear unbiased predictor under the mixed effects model and that is design-consistent is presented. An estimator for the design variance of the regression estimator is developed.;A weighted regression estimator that builds upon the approximate conditional inclusion probabilities of the sample elements is developed. The sample weights of the weighted regression estimator are positive with high probability.;It is demonstrated that the strategy of the best linear unbiased predictor with a balanced sample is similar to the strategy of a regression estimator with a random sample. A balanced sample constructed by restricted random sampling and two stratified random sampling designs, both using the regression estimator, are compared for the problem of estimating the population distribution function. The regression estimator with a stratified sample gives smaller bias and smaller mean squared error than the best linear unbiased predictor with a restricted random sample for almost all points on the distribution function.