In this paper influence curves for censored data estimators are considered. The influence curve of the Kaplan-Meier estimate of survival time is calculated, and it is shown that the influence curve provides an alternative derivation of the asymptotic variance of this estimate. A chain rule for influence curves is established and is used to calculate the influence curves of censored data estimators that are functions of the Kaplan-Meier estimate. The robustness properties and asymptotic variances of these estimates follow directly. Some examples of this approach to calculating variances are given. In particular, it is shown how the theory developed for $M$-estimation and $L$-estimation can be extended to the censored data case. The necessary differentiability conditions are verified in the appendix.