Title: $\ell_\infty$-Robustness and Beyond: Unleashing Efficient Adversarial Training. Abstract: Neural networks are vulnerable to adversarial attacks: adding well-crafted, imperceptible perturbations to their input can modify their output. Adversarial training is one of the most effective approaches in training robust models against such attacks. However, it is much slower than vanilla training of neural networks since it needs to construct adversarial examples for the entire training data at every iteration, which has hampered its effectiveness. Recently, Fast Adversarial Training was proposed that can obtain robust models within minutes. However, the reasons behind its success are not fully understood, and more importantly, it can only train robust models for $\ell_\infty$-bounded attacks as it uses FGSM during training. In this paper, by leveraging the theory of coreset selection we show how selecting a small subset of training data provides a more principled approach towards reducing the time complexity of robust training. Unlike Fast Adversarial Training, our approach can be adapted to a wide variety of training objectives, including TRADES, $\ell_p$-PGD, and Perceptual Adversarial Training. Our experimental results indicate that using coreset selection, one can train robust models 2-3 times faster while maintaining the clean and robust accuracy almost intact.