We first propose the single-index hazards model for right censored survival data. As an extension of the Cox model, this model allows nonparametric modeling of covariate effects in a parsimonious way via a single-index. In addition, the relative importance of covariates can be assessed via this model. We consider the conventional profile-kernel method based on the local likelihood for model estimation. It is shown that this method may give consistent estimation under certain restrictive conditions, but in general it can yield biased estimation. Simulation studies are conducted to demonstrate the bias phenomena. The existence and nature of the failure of this commonly used approach is somewhat surprising. The interpretation of covariate effects in the aforementioned single-index hazards model is difficult. Thus, we further propose the partly proportional single-index hazards model in which the effect of covariates of primary interest is represented by the regression parameter while "nuisance" covariates can have nonparametric effect on the survival time. We again consider the conventional profile-kernel method and it leads to biased estimation as well. A bias correction method is then proposed and the corrected profile local likelihood estimators are shown to be consistent, asymptotically normal and semiparametrically efficient. We evaluate the finite-sample properties of our estimators through simulation studies and illustrate the proposed model and method with an application to a dataset from the Multicenter AIDS Cohort Study (MACS). Besides the profile-kernel method, we also study the profile stratified likelihood method based on stratification of the single-index. In the single-index hazards model, this method may give consistent estimation under the restrictive "independent censoring" condition, but in general it can yield biased estimation. Simulation studies are conducted to demonstrate the situations in which the bias phenomena do (or do not) exist; In the partly proportional single-index hazards model, we demonstrate numerically the existence of the bias and then propose a bias correction method. The estimators from the corrected profile stratified likelihood method are shown to be consistent. Their finite-sample properties are evaluated through simulation studies. The corrected profile stratified method is applied to the aforementioned MACS study for illustration.