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In this article we give a general definition of residuals for regression models with independent responses. Our definition produces residuals that are exactly normal, apart from sampling variability in the estimated parameters, by inverting the fitted distribution function for each response value and finding the equivalent standard normal quantile. Our definition includes some randomization to achieve continuous residuals when the response variable is discrete. Quantile residuals are easily computed in computer packages such as SAS, S-Plus, GLIM, or LispStat, and allow residual analyses to be carried out in many commonly occurring situations in which the customary definitions of residuals fail. Quantile residuals are applied in this article to three example data sets.