Effective screening methods are crucial to the analysis of big biomedical data. The popular sure independence screening relies on restricted assumptions such as the partial faithfulness condition, e.g, the partial correlations between outcome and covariates can be inferred from their marginal correlations. However, such a restrictive assumption is often violated, as the marginal effects of predictors may differ from their joint effects, especially when the covariates are correlated. We propose a covariance-insured screening (CIS) framework that utilizes the dependence among covariates and identify important features that are likely to be missed by marginal screening procedures. The proposed framework encompasses linear regression models, generalized linear regression models, survival models, and classification of multi-level outcomes. The methods will be evaluated by simulations and real data examples.