This paper analyzes the linear regression model y = x + with a conditional median assumption Med( j z) = 0 where z is a vector of exogenous random variables. Added complication arise due to the censoring of the outcome y. We treat the censored model as a model with interval-observed outcomes thus obtaining an incomplete model with inequality restrictions on conditional median regressions. This allows us to use the estimator introduced by Manski and Tamer (2000) to analyze the information contained in these inequality restrictions. We give identication conditions in the absence of censoring and introduce a p N-consistent estimator based on the minimum distance method. We then give suÆcient conditions for global identication of with censored y and endogenous x. In the case of interval data on y and endogenous x, we provide a set-consistent estimator that is based on a modied minimum distance method. In the case where we have point identication, we show that the estimator is p N-normal and derive its asymptotic distribution with a feasible asymptotic variance. A Montecarlo analysis illustrates our estimator. We thank Bo Honore for comments and the Econometrics Research Program at Princeton for support.