"The Structure of Classical Diffeomorphism Groups" is a comprehensive monograph that explores the structure and properties of diffeomorphism groups of smooth manifolds. Based on the work of Anderson, Chernavski, Kirby, and Edwards from the 1960s, it investigates Smale's hypothesis that the group of cr-diffeomorphisms with compact support that are isotopic to the identity is also a simple group. The author, Augustin Banyaga, provides a detailed proof of this thesis and highlights the deep connections between the local homology of diffeomorphism groups and the homology of the Haefliger classification space for foliations. The monograph also addresses the so-called Thurston tricks, which demonstrate how perfection in certain diffeomorphism groups leads to simplicity. These theoretical concepts are complemented by the application of Epstein's theory to contact diffeomorphisms, offering profound insights into modern differential geometry.