Social scientists typically devote considerable effort to reducing measurement error during data collection and then ignore the issue during data analysis. Although many statistical methods have been proposed for reducing measurement error-induced biases, few have been widely used because of implausible assumptions, high levels of model dependence, difficult computation, or inapplicability with multiple mismeasured variables. We develop an easy-to-use alternative that generalizes the popular multiple imputation (MI) framework by treating missing data problems as a special case of extreme measurement error and correcting for both. Like MI, the proposed "multiple overimputation" (MO) framework is a simple two-step procedure. First, multiple (≈5) completed copies of the data set are created where cells measured without error are held constant, those missing are imputed from the distribution of predicted values, and cells (or entire variables) with measurement error are "overimputed," that is imputed from a predictive distribution with observation-level priors defined by the mismeasured values and available external information, if any. In the second step, analysts can then run whatever statistical method they would have run on each of the overimputed data sets as if there had been no missingness or measurement error; the results are then combined via a simple procedure. We also (will) offer open source software that implements all the methods described herein.