The additive hazards regression model relates the conditional hazard function of the failure time linearly to the covariates. This formulation complements the familiar proportional hazards model in that it describes the association between the covariates and failure time in terms of the risk difference rather than the risk ratio. In this paper, we provide a closed-form semiparametric estimator for the (vector-valued) regression parameter of the additive hazards model with right-censored data, which is consistent and asymptotically normal with a simple variance estimator. We also demonstrate how the additive hazards framework can be used effectively to incorporate frailty and to handle interval-censored data, the resulting semiparametric inference procedures being much simpler than their counterparts under the proportional hazards framework.