Understanding the great performances of deep neural networks is a very active direction of research with contributions coming from a wide variety of fields. The statistical mechanics of learning is a theoretical framework dating back to the 80s studying learning problems from a physicist viewpoint and using tools from the physics of disordered systems. In this talk, I will first go over this traditional framework, which relies on the teacher-student scenario, bayesian analysis and mean-field approximations. Then I will discuss some recent advances in the corresponding analysis of modern deep neural network, and highlight remaining challenges.