. We begin by analyzing the local adaptation properties of wavelet-based curve estimators. It is argued that while wavelet methods enjoy outstanding adaptability in terms of the manner in which they capture irregular episodes in a curve, they are not nearly as adaptive when considered from the viewpoint of tracking more subtle changes in a smooth function. We point out that while this problem may be remedied by modifying wavelet estimators, simple modifications are typically not sufficient to properly achieve adaptive smoothing of a relatively highly differentiable function. In that case, local changes to the primary level of resolution of the wavelet transform are required. While such an approach is feasible, it is not an attractive proposition on either practical or aesthetic grounds. Motivated by this difficulty, we develop local versions of familiar smoothing methods, such as cross-validation and smoothed cross-validation, in the contexts of density estimation and regression. It is..