State-of-the-art results on image recognition tasks are achieved using over-parameterized learning algorithms that (nearly) perfectly fit the training set and are known to fit well even random labels. This tendency to memorize the labels of the training data is not explained by existing theoretical analyses. Memorization of the training data also presents significant privacy risks when the training data contains sensitive personal information and thus it is important to understand whether such memorization is necessary for accurate learning. We provide a simple conceptual explanation and a theoretical model demonstrating that for natural data distributions memorization of labels is necessary for achieving close-to-optimal generalization error. The model is motivated and supported by the results of several recent empirical works. In our model, data is sampled from a mixture of subpopulations and the frequencies of these subpopulations are chosen from some prior. The model allows to quantify the effect of not fitting the training data on the generalization performance of the learned classifier and demonstrates that memorization is necessary whenever frequencies are long-tailed. Image and text data are known to follow such distributions and therefore our results establish a formal link between these empirical phenomena. Our results also have concrete implications for the cost of ensuring differential privacy in learning.