Today's machine learning is all about "large dimensional (p) and numerous data (n)", in sharp contrast with yesterday's "small p, large n" asymptotics paradigm. In this talk, we will show that moving from a "small p, large n" to a "large p, large n" setting has major consequences that disrupt one's mathematical intuitions. To this end we introduce basic notions of random matrix theory and provide new intuitions adapted to the "large p, large n" regime. We will then show how these notions are used to derive new results for the performance understanding and improvement of popular machine learning methods, such as SVMs, kernels, neural nets, etc.