Summary: The Dantzig variable selector has recently emerged as a powerful tool for fitting regularized regression models. To our knowledge, most work involving the Dantzig selector has been performed with fully-observed response variables. This paper proposes a new class of adaptive Dantzig variable selectors for linear regression models when the response variable is subject to right censoring. This is motivated by a clinical study to identify genes predictive of event-free survival in newly diagnosed multiple myeloma patients. Under some mild conditions, we establish the theoretical properties of our procedures, including consistency in model selection and the optimal efficiency of estimation. The practical utility of the proposed adaptive Dantzig selectors is verified via extensive simulations. We apply our new methods to the aforementioned myeloma clinical trial and identify important predictive genes.