Any diagram group over a +nite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satis+es the Baum–Connes conjecture. Any diagram group over a +nite presentation ...