The Magic Wand Theorem of A. Eskin and M. Mirzakhani

This is the English translation of a short note published by Gazette des Math\'ematiciens. The author was asked to present the recent work of Alex Eskin and of Maryam Mirzakhani, arXiv:1302.3320, and their joint work with Amir Mohammadi, arXiv:1305.3015, on SL(2,R)-action on the moduli space of Abelian differentials. This is a popularization paper with no proofs. Its only purpose is to suggest some motivations, and to outline possible applications considering billiards in polygons as an example.