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 timeYgivenY≥ τ can be estimated nonparametrically, where τ is the lower boundary of the support of the left-truncation variableT. 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 variableC. In this article we first consider nonparametric estimation of trimmed functionals of the conditional distribution ofY, 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 variableY. Asymptotic normality of M estimators of the regression parameters derived from this approach is established, and the result is used to construct confidence regions for the regression parameters. We also apply our methods of nonparametric estimation, correlation analysis, and curve fitting for left-truncated and right-censored data to analyze transfusion-induced AIDS data, and present a simulation study comparing our approach with another kind of M estimators for regression analysis in the presence of left truncation and right censoring.