Machine learning models trained on data from the outside world can be corrupted by data poisoning attacks that inject malicious points into the models' training sets. A common defense against these attacks is data sanitization: first filter out anomalous training points before training the model. In this paper, we develop three attacks that can bypass a broad range of common data sanitization defenses, including anomaly detectors based on nearest neighbors, training loss, and singular-value decomposition. By adding just 3% poisoned data, our attacks successfully increase test error on the Enron spam detection dataset from 3% to 24% and on the IMDB sentiment classification dataset from 12% to 29%. In contrast, existing attacks which do not explicitly account for these data sanitization defenses are defeated by them. Our attacks are based on two ideas: (i) we coordinate our attacks to place poisoned points near one another, and (ii) we formulate each attack as a constrained optimization problem, with constraints designed to ensure that the poisoned points evade detection. As this optimization involves solving an expensive bilevel problem, our three attacks correspond to different ways of approximating this problem, based on influence functions; minimax duality; and the Karush-Kuhn-Tucker (KKT) conditions. Our results underscore the need to develop more robust defenses against data poisoning attacks.