Bias due to unobserved confounding can seldom be ruled out with certainty when estimating the causal effect of a nonrandomized treatment. The instrumental variable (IV) design offers, under certain assumptions, the opportunity to tame confounding bias, without directly observing all confounders. The IV approach is very well developed in the context of linear regression and also for certain generalized linear models with a nonlinear link function. However, IV methods are not as well developed for regression analysis with a censored survival outcome. In this article, we develop the IV approach for regression analysis in a survival context, primarily under an additive hazards model, for which we describe 2 simple methods for estimating causal effects. The first method is a straightforward 2-stage regression approach analogous to 2-stage least squares commonly used for IV analysis in linear regression. In this approach, the fitted value from a first-stage regression of the exposure on the IV is entered in place of the exposure in the second-stage hazard model to recover a valid estimate of the treatment effect of interest. The second method is a so-called control function approach, which entails adding to the additive hazards outcome model, the residual from a first-stage regression of the exposure on the IV. Formal conditions are given justifying each strategy, and the methods are illustrated in a novel application to a Mendelian randomization study to evaluate the effect of diabetes on mortality using data from the Health and Retirement Study. We also establish that analogous strategies can also be used under a proportional hazards model specification, provided the outcome is rare over the entire follow-up.