In data sets with many more features than observations, independent screening based on all univariate regression models leads to a computationally convenient variable selection method. Recent efforts have shown that, in the case of generalized linear models, independent screening may suffice to capture all relevant features with high probability, even in ultrahigh dimension. It is unclear whether this formal sure screening property is attainable when the response is a right-censored survival time. We propose a computationally very efficient independent screening method for survival data which can be viewed as the natural survival equivalent of correlation screening. We state conditions under which the method admits the sure screening property within a class of single-index hazard rate models with ultrahigh dimensional features and describe the generally detrimental effect of censoring on performance. An iterative variant of the method is also described which combines screening with penalized regression to handle more complex feature covariance structures. The methodology is evaluated through simulation studies and through application to a real gene expression data set.