Randomly censored survival data appear in a wide variety of applications in which the time until the occurrence

of a certain event is not completely observable. In this paper, we assume that the statistician observes a

possibly censored survival time along with a censoring indicator. In this setting,we study a class of M-estimators

with a bounded influence function, in the spirit of the infinitesimal approach to robustness. We outline the

main asymptotic properties of the robust M-estimators and characterize the optimal B-robust estimator according

to two possible measures of sensitivity. Building on these results, we define robust testing procedures which

are natural counterparts to the classical Wald, score, and likelihood ratio tests. The empirical performance of

our robust estimators and tests is assessed in two extensive simulation studies. An application to data from a

well-known medical study on head and neck cancer is also presented.