The 2019 course by Stéphane Mallat, Professor at the Collège de France, continues the trend from 2018 by emphasizing 'Why does it work?' This implies that while there are many courses that focus on 'How' to implement various architectures of convolutional neural networks (CNNs), the 'Why' falls within the realm of fundamental research. The course first describes how the ideas of neural networks originated, their different types of applications, and the mathematical questions underlying their success. The demonstration of the Universality theorem of networks with 1 hidden layer is central to the course, as well as its practical inefficiency in the face of the curse of dimensionality. The end of the course covers notions related to optimization.