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This paper describes the R add-on package BradleyTerry, which facilitates the specification and fitting of Bradley-Terry logit models to pair-comparison data. Included are the standard "unstructured" Bradley-Terry model, structured versions in which the parameters are related through a linear predictor to explanatory variables, and the possibility of an order or "home advantage" effect. Model fitting is either by maximum likelihood or by bias-reduced maximum likelihood in which the first-order asymptotic bias of parameter estimates is eliminated. Also provided are a simple and efficient approach to handling missing covariate data, and suitably-defined residuals for diagnostic checking of the linear predictor; these are new methodological contributions which will be discussed in greater detail elsewhere.