As a computationally fast and working efficient tool, sure independence screening has received much attention in solving ultrahigh dimensional problems. This paper contributes two robust sure screening approaches that simultaneously take into account heteroscedasticity, outliers, heavy-tailed distribution, continuous or discrete response, and confounding effect, from the perspective of model-free. First, we define a robust correlation measure only using two random indicators, and introduce a screener using that correlation. Second, we propose a robust partial correlation-based screening approach when an exposure variable is available. To remove the confounding effect of the exposure on both response and each covariate, we use a nonparametric regression with some specified loss function. More specifically, a robust correlation-based screening method (RC-SIS) and a robust partial correlation-based screening framework (RPC-SIS) including two concrete screeners: RPC-SIS(L2) and RPC-SIS(L1), are formed. Third, we establish sure screening properties of RC-SIS for which the response variable can be either continuous or discrete, as well as those of RPC-SIS(L2) and RPC-SIS(L1) under some regularity conditions. Our approaches are essentially nonparametric, and perform robustly for both the response and the covariates. Finally, extensive simulation studies and two applications are carried out to demonstrate the superiority of our proposed approaches.