In this work, we study the connection between regularization and robustness of deep neural networks by viewing them as elements of a reproducing kernel Hilbert space (RKHS) of functions and by regularizing them using the RKHS norm. Even though this norm cannot be computed, we consider various approximations based on upper and lower bounds. These approximations lead to new strategies for regularization, but also to existing ones such as spectral norm penalties or constraints, gradient penalties, or adversarial training. Besides, the kernel framework allows us to obtain margin-based bounds on adversarial generalization. We show that our new algorithms lead to empirical benefits for learning on small datasets and learning adversarially robust models. We also discuss implications of our regularization framework for learning implicit generative models.