2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/136372We report on recent progress in the program to classify separable amenable C*-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.19 pages, to appear in Bull. AMS, April 2008Operator AlgebrasFunctional Analysis46L35; 46L80text