  Driven by massive amounts of data and important advances in computational
resources, new deep learning systems have achieved outstanding results in a
large spectrum of applications. Nevertheless, our current theoretical
understanding on the mathematical foundations of deep learning lags far behind
its empirical success. Towards solving the vulnerability of neural networks,
however, the field of adversarial robustness has recently become one of the
main sources of explanations of our deep models. In this article, we provide an
in-depth review of the field of adversarial robustness in deep learning, and
give a self-contained introduction to its main notions. But, in contrast to the
mainstream pessimistic perspective of adversarial robustness, we focus on the
main positive aspects that it entails. We highlight the intuitive connection
between adversarial examples and the geometry of deep neural networks, and
eventually explore how the geometric study of adversarial examples can serve as
a powerful tool to understand deep learning. Furthermore, we demonstrate the
broad applicability of adversarial robustness, providing an overview of the
main emerging applications of adversarial robustness beyond security. The goal
of this article is to provide readers with a set of new perspectives to
understand deep learning, and to supply them with intuitive tools and insights
on how to use adversarial robustness to improve it.
