  The purpose of this article is to review the achievements made in the last
few years towards the understanding of the reasons behind the success and
subtleties of neural network-based machine learning. In the tradition of good
old applied mathematics, we will not only give attention to rigorous
mathematical results, but also the insight we have gained from careful
numerical experiments as well as the analysis of simplified models. Along the
way, we also list the open problems which we believe to be the most important
topics for further study. This is not a complete overview over this quickly
moving field, but we hope to provide a perspective which may be helpful
especially to new researchers in the area.
