Existing methods for the analysis of data involving assay data subject to measurement error are deficient. In particular, classical calibration methods have been shown to yield invalid inferences unless the measurement error is small. Regression calibration, a form of conditional mean imputation, has better properties, but is not well suited to adjusting for heteroscedastic measurement error. Bayesian multiple imputation is less common for measurement error problems than for missing data, but we argue that it represents an attractive option for measurement error, providing superior inferences to existing methods and a convenient way of adjusting for measurement error using simple complete-data methods and multiple imputation combining rules. It also provides a convenient approach to limit of quantification issues, another area where current approaches are in our view deficient. We review some recent work that develops multiple imputation methods for assay data, focusing particularly on three key aspects: internal versus external calibration designs, the role of the nondifferential measurement error assumption in these designs, and heteroscedastic measurement error. Future research topics are outlined. Here the AMS Subject Classification http://www.ams.org/msc/; Existing methods for the analysis of data involving assay data subject to measurement error are deficient. In particular, classical calibration methods have been shown to yield invalid inferences unless the measurement error is small. Regression calibration, a form of conditional mean imputation, has better properties, but is not well suited to adjusting for heteroscedastic measurement error. Bayesian multiple imputation is less common for measurement error problems than for missing data, but we argue that it represents an attractive option for measurement error, providing superior inferences to existing methods and a convenient way of adjusting for measurement error using simple complete-data methods and multiple imputation combining rules. It also provides a convenient approach to limit of quantification issues, another area where current approaches are in our view deficient. We review some recent work that develops multiple imputation methods for assay data, focusing particularly on three key aspects: internal versus external calibration designs, the role of the nondifferential measurement error assumption in these designs, and heteroscedastic measurement error. Future research topics are outlined. Here the AMS Subject Classification http://www.ams.org/msc/