Artificial intelligence is currently leading to one breakthrough after the other, both in public life with, for instance, autonomous driving and speech recognition, and in the sciences in areas such as medical imaging or molecular dynamics. However, one current major drawback is the lack of reliability of such methodologies. In this talk we will take a mathematical viewpoint towards this problem, showing the power of such approaches to reliability. We will first provide an introduction to this vibrant research area, focussing specifically on deep neural networks. We will then survey recent advances, in particular, concerning generalization guarantees and explainability. Finally, we will discuss the fundamental limitations of deep neural networks and related approaches in terms of computability, which seriously affects their reliability. Part of the LMS/IMA Joint Meeting on 'The Mathematical Foundations' of AI, which took place on Friday 13 October 2023 at De Morgan House, London and online via Zoom. ========== The London Mathematical Society has, since 1865, been the UK's learned society for the advancement, dissemination and promotion of mathematical knowledge. Our mission is to advance mathematics through our members and the broader scientific community worldwide. For further information: ► Website: https://www.lms.ac.uk/ ► Events: https://www.lms.ac.uk/events ► Grants and Prizes: https://www.lms.ac.uk/grants-prizes ► Publications: https://www.lms.ac.uk/publications ► Membership: https://www.lms.ac.uk/membership Follow us: ► Twitter: https://twitter.com/LondMathSoc ► Facebook: https://www.facebook.com/londonmathematicalsociety ► LinkedIn: https://www.linkedin.com/company/the-london-mathematical-society/ ► Youtube: @LondonMathematicalSociety