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To accommodate the heterogeneity that is often present in ultrahigh-dimensional data, we propose a conditional quantile screening method, which enables us to select features that contribute to the conditional quantile of the response given the covariates. The method can naturally handle censored data by incorporating a weighting scheme through redistribution of the mass to the right; moreover, it is invariant to monotone transformation of the response and requires substantially weaker conditions than do alternative methods. We establish sure independent screening properties for both the complete and the censored response cases. We also conduct simulations to evaluate the finite-sample performance of the proposed method, and compare it with existing approaches.