A common problem in spatial statistics is to predict a random field f at some spatial location t(0) using observations f(t(1)),..., f(t(n)) at t(1),..., t(n) epsilon IRd. Recent work by Kaufman et al. and Furrer et al. studies the use of tapering for reducing the computational burden associated with likelihood-based estimation and prediction in large spatial datasets. Unfortunately, highly irregular observation locations can present problems for stationary tapers. In particular, there can exist local neighborhoods with too few observations for sufficient accuracy, while others have too many for computational tractability. In this article, we show how to generate nonstationaty covariance tapers T(s, t) such that the number of observations in {t : T(s, t) > 0} is approximately a constant function of s. This ensures that tapering neighborhoods do not have too many points to cause computational problems but simultaneously have enough local points for accurate prediction. We focus specifically on tapering in two dimensions where quasi-conformal theory can be used. Supplementary materials for the article are available online.