Large, overparameterized models such as neural networks are now the workhorses of modern machine learning. These models are often trained to near-zero error on noisy datasets and simultaneously generalize well to unseen data, in contrast to the textbook intuition regarding the perils of overfitting. At the same time, near-perfect data-fitting can have severe issues in the context of robustness, privacy, and fairness. Classical theoretical frameworks provide little guidance for navigating these questions due to overparameterization. It is thus crucial to develop new intuition regarding overfitting and generalization that are reflective of these empirical observations. In this tutorial, we discuss recent work in the learning theory literature that provides theoretical insights into these phenomena.