Abstract: In many applications involving follow-up studies, individuals' lifetimes may be subject to both left and right censoring. In this paper we consider regressing the median of the lifetime variable or a transformation thereof on the corresponding covariates when the lifetimes are subject to both left and right censoring and the values of both censoring variables are always observable. A semiparametric inferential procedure is proposed and the asymptotic properties of the proposed method are discussed. We also propose an alternative empirical likelihood based inferential procedure for the regression coefficients vector. Moreover, we make some discussion on classical doubly censored data whose censoring variables cannot always be observed. The proposed methods are illustrated by some simulation studies.