In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers. To address this, we introduce a new meta-algorithm that can take in a base learner such as least squares or stochastic gradient descent, and harden the learner to be resistant to outliers. Our method, Sever, possesses strong theoretical guarantees yet is also highly scalable -- beyond running the base learner itself, it only requires computing the top singular vector of a certain $n \times d$ matrix. We apply Sever on a drug design dataset and a spam classification dataset, and find that in both cases it has substantially greater robustness than several baselines. On the spam dataset, with $1\%$ corruptions, we achieved $7.4\%$ test error, compared to $13.4\%-20.5\%$ for the baselines, and $3\%$ error on the uncorrupted dataset. Similarly, on the drug design dataset, with $10\%$ corruptions, we achieved $1.42$ mean-squared error test error, compared to $1.51$-$2.33$ for the baselines, and $1.23$ error on the uncorrupted dataset.