Machine learning is vulnerable to a wide variety of different attacks. It is now well understood that by changing the underlying data distribution, an adversary can poison the model trained with it or introduce backdoors. In this paper we present a novel class of training-time attacks that require no changes to the underlying model dataset or architecture, but instead only change the order in which data are supplied to the model. In particular, an attacker can disrupt the integrity and availability of a model by simply reordering training batches, with no knowledge about either the model or the dataset. Indeed, the attacks presented here are not specific to the model or dataset, but rather target the stochastic nature of modern learning procedures. We extensively evaluate our attacks to find that the adversary can disrupt model training and even introduce backdoors. For integrity we find that the attacker can either stop the model from learning, or poison it to learn behaviours specified by the attacker. For availability we find that a single adversarially-ordered epoch can be enough to slow down model learning, or even to reset all of the learning progress. Such attacks have a long-term impact in that they decrease model performance hundreds of epochs after the attack took place. Reordering is a very powerful adversarial paradigm in that it removes the assumption that an adversary must inject adversarial data points or perturbations to perform training-time attacks. It reminds us that stochastic gradient descent relies on the assumption that data are sampled at random. If this randomness is compromised, then all bets are off.