This paper proposes alternatives to maximum likelihood estimation of the censored and truncated regression models (known to economists as "Tobit" models). The proposed estimators are based upon symmetric censoring or truncation of the upper tail of the distribution of the dependent variable. Unlike methods based on the assumption of identically distributed Gaussian errors, the estimators are semiparametric, in the sense that they are consistent and asymptotically normal for a wide class of (symmetric) error distributions with heteroskedasticity of unknown form. The paper gives the regularity conditions and proofs of these large sample properties, demonstrates how to construct consistent estimators of the asymptotic covariance matrices, and presents the results of a simulation study for the censored case. Extensions and limitations of the approach are also considered.