Andrea Montanari received a Laurea degree in Physics in 1997, and a Ph. D. in Theoretical Physics in 2001 (both from Scuola Normale Superiore in Pisa, Italy). He has been post-doctoral fellow at Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (LPTENS), Paris, France, and the Mathematical Sciences Research Institute, Berkeley, USA. Since 2002 he is Chargé de Recherche (with Centre National de la Recherche Scientifique, CNRS) at LPTENS. In September 2006 he joined Stanford University as a faculty, and since 2015 he is Full Professor in the Departments of Electrical Engineering and Statistics. He was co-awarded the ACM SIGMETRICS best paper award in 2008. He received the CNRS bronze medal for theoretical physics in 2006, the National Science Foundation CAREER award in 2008, the Okawa Foundation Research Grant in 2013, and the Applied Probability Society Best Publication Award in 2015. He is an Information Theory Society distinguished lecturer for 2015-2016. In 2016 he received the James L. Massey Research & Teaching Award of the Information Theory Society for young scholars, and in 2017 was elevated to IEEE Fellow. In 2018 he was an invited sectional speaker at the International Congress of Mathematicians. He is an invited IMS Medallion lecturer for the 2020 Bernoulli-IMS World Congress. Abstract: The last fifteen years have witnessed dramatic breakthroughs in machine learning. This progress was crucially driven by engineering advances: greater computing power and larger availability of training data. Not only the collection of methods that emerged from this revolution are not well understood mathematically, but they actually appear to defy traditional mathematical theories of machine learning. Future developments and applications, especially within the sciences, will require to understand better the underlying mathematical principles. I will try to provide a gentle introduction to the subject, and a peek into some recent advances towards addressing these challenges. The colloquium will be via Zoom. Pre-registration is required at the following Zoom link: https://zoom.us/webinar/register/WN_YK6VZDvDT3WdBljtsqhHWQ. You will then receive a confirmation email. The colloquium will be livestreamed from the ICTP website. All are welcome to join.