The extreme fragility of deep neural networks, when presented with tiny perturbations in their inputs, was independently discovered by several research groups in 2013. However, despite enormous effort, these adversarial examples remained a counterintuitive phenomenon with no simple testable explanation. In this paper, we introduce a new conceptual framework for how the decision boundary between classes evolves during training, which we call the {\em Dimpled Manifold Model}. In particular, we demonstrate that training is divided into two distinct phases. The first phase is a (typically fast) clinging process in which the initially randomly oriented decision boundary gets very close to the low dimensional image manifold, which contains all the training examples. Next, there is a (typically slow) dimpling phase which creates shallow bulges in the decision boundary that move it to the correct side of the training examples. This framework provides a simple explanation for why adversarial examples exist, why their perturbations have such tiny norms, and why they look like random noise rather than like the target class. This explanation is also used to show that a network that was adversarially trained with incorrectly labeled images might still correctly classify most test images, and to show that the main effect of adversarial training is just to deepen the generated dimples in the decision boundary. Finally, we discuss and demonstrate the very different properties of on-manifold and off-manifold adversarial perturbations. We describe the results of numerous experiments which strongly support this new model, using both low dimensional synthetic datasets and high dimensional natural datasets.