Predictor screening rules, which discard predictors before fitting a model, have had considerable impact on the speed with which sparse regression problems, such as the lasso, can be solved. In this paper we present a new screening rule for solving the lasso path: the Hessian Screening Rule. The rule uses second-order information from the model to provide both effective screening, particularly in the case of high correlation, as well as accurate warm starts. The proposed rule outperforms all alternatives we study on simulated data sets with both low and high correlation for $\ell_1$-regularized least-squares (the lasso) and logistic regression. It also performs best in general on the real data sets that we examine.