In many prospective and retrospective studies, survival data are subject to left truncation in addition to the usual right censoring. For left-truncated data without covariates, only the conditional distribution of the survival time Y given Y ≥ τ can be estimated nonparametrically, where τ is the lower boundary of the support of the left-truncation variable T. If the data are also right censored, then the conditional distribution can be consistently estimated only at points not larger than τ*, where τ* is the upper boundary of the support of the right-censoring variable C. In this article we first consider nonparametric estimation of trimmed functionals of the conditional distribution of Y, with the trimming inside the observable range between τ and τ*. We then extend the approach to regression analysis and curve fitting in the presence of left truncation and right censoring on the response variable Y. Asymptotic normality of M estimators of the regression parameters derived from this approach is...