Summary We study accelerated failure time models in which the survivor function of the additive error term is log-concave. The log-concavity assumption covers large families of commonly used distributions and also represents the aging or wear-out phenomenon of the baseline duration. For right-censored failure time data, we construct semiparametric maximum likelihood estimates of the finite-dimensional parameter and establish the large sample properties. The shape restriction is incorporated via a nonparametric maximum likelihood estimator of the hazard function. Our approach guarantees the uniqueness of a global solution for the estimating equations and delivers semiparametric efficient estimates. Simulation studies and empirical applications demonstrate the usefulness of our method.