Adversarial attacks alter NLP model predictions by perturbing test-time inputs. However, it is much less understood whether, and how, predictions can be manipulated with small, concealed changes to the training data. In this work, we develop a new data poisoning attack that allows an adversary to control model predictions whenever a desired trigger phrase is present in the input. For instance, we insert 50 poison examples into a sentiment model's training set that causes the model to frequently predict Positive whenever the input contains James Bond''. Crucially, we craft these poison examples using a gradient-based procedure so that they do not mention the trigger phrase. We also apply our poison attack to language modeling (Apple iPhone'' triggers negative generations) and machine translation (iced coffee'' mistranslated as hot coffee''). We conclude by proposing three defenses that can mitigate our attack at some cost in prediction accuracy or extra human annotation.