Adversarial training (AT) has become a popular choice for training robust networks. However, it tends to sacrifice clean accuracy heavily in favor of robustness and suffers from a large generalization error. To address these concerns, we propose Smooth Adversarial Training (SAT), guided by our analysis on the eigenspectrum of the loss Hessian. We find that curriculum learning, a scheme that emphasizes on starting "easy'' and gradually ramping up on the "difficulty'' of training, smooths the adversarial loss landscape for a suitably chosen difficulty metric. We present a general formulation for curriculum learning in the adversarial setting and propose two difficulty metrics based on the maximal Hessian eigenvalue (H-SAT) and the softmax probability (P-SA). We demonstrate that SAT stabilizes network training even for a large perturbation norm and allows the network to operate at a better clean accuracy versus robustness trade-off curve compared to AT. This leads to a significant improvement in both clean accuracy and robustness compared to AT, TRADES, and other baselines. To highlight a few results, our best model improves normal and robust accuracy by 6% and 1% on CIFAR-100 compared to AT, respectively. On Imagenette, a ten-class subset of ImageNet, our model outperforms AT by 23% and 3% on normal and robust accuracy respectively.