In many medical and scientific settings, the choice of treatment or intervention may be determined by a covariate threshold. For example, elderly men may receive more thorough diagnosis if their prostate-specific antigen (PSA) level is high. In these cases, the causal treatment effect is often of great interest, especially when there is a lack of evidence from randomized clinical trials. From the social science literature, a class of methods known as regression discontinuity (RD) designs can be used to estimate the treatment effect in this situation. Under certain assumptions, such an estimand enjoys a causal interpretation. We show how to estimate causal effects under the regression discontinuity design for censored data. The proposed estimation procedure employs a class of censoring unbiased transformations that includes inverse probability censored weighting and doubly robust transformation schemes. Simulation studies are used to evaluate the finite-sample properties of the proposed estimator. We also illustrate the proposed method by evaluating the causal effect of PSA-dependent screening strategies.