  We currently witness the spectacular success of artificial intelligence in
both science and public life. However, the development of a rigorous
mathematical foundation is still at an early stage. In this survey article,
which is based on an invited lecture at the International Congress of
Mathematicians 2022, we will in particular focus on the current "workhorse" of
artificial intelligence, namely deep neural networks. We will present the main
theoretical directions along with several exemplary results and discuss key
open problems.
