Stochastic curtailment is a valuable tool in monitoring long-term medical studies. Under this approach, one calculates the conditional power, which is the probability of rejecting the null hypothesis at the scheduled end of the study given the existing data at the interim analysis, along with certain speculation about the future data. The conditional power may be used to aid the decision to terminate a study prematurely or to extend a study beyond its originally planned duration. This article provides a formal and systematic investigation into the use of stochastic curtailment in the context of censored survival data. To enhance generality, we introduce a broad class of statistics that includes two-sample weighted log-rank statistics, as well as the partial likelihood score statistic for testing treatment difference with covariate adjustment under the proportional hazards model. We establish the weak convergence under both the null hypothesis and contiguous alternatives for this class of statistics when calculated repeatedly over the calendar time (i.e., time of interim analysis). Further, we derive the conditional distributions of these statistics calculated at the end of the study given all the data collected up to the interim look or given the statistics calculated at the interim look, and provide analytic expressions for the corresponding conditional powers. These results enable us to address several subtle issues involved in the definition and implementation of conditional power for censored survival data, especially when there is staggered patient entry with a potential time trend in the survival distribution, when the Gehan-type weight function is used, or when treatment is not independent of covariates. For randomized clinical trials, we show that very simple formulas can be used to calculate the conditional powers of the unweighted log-rank test (with or without covariate adjustment) under both the null and alternative hypotheses. Simulation studies demonstrate that the conditional powers for survival studies can be accurately evaluated through the proposed formulas even when the sample size is small. An illustration with data taken from a colon cancer study is provided.