A technique is suggested for reducing the order of bias of kernel estimators by weighting the contributions that different data values make to the estimator. The method is developed initially in the context of density estimation, where, unlike the “variable kernel” method proposed by Abramson, our approach does not involve using different bandwidths at different data values. Rather, it is a weighted-bootstrap version of the standard uniform-bootstrap method that is used to construct traditional kernel density estimators. The reduction in bias is achieved by biasing the bootstrap appropriately, in a global rather than local way. Our technique has a variety of different forms, each of which reduces the order of bias from the square to the fourth power of bandwidth, but does not alter the order of variance. It has immediate application to nonparametric regression, where it allows bias to be reduced without prejudicing the one sign of an estimator. ; Peter Hall and Berwin A. Turlach