Abstract:Alternative titles of this paper would have been ``Index theory without index'' or ``The BaumConnes conjecture without Baum.'' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include WeylMoyal quantization on manifolds, C-algebras of Lie groups and Lie groupoids, and the E-theoretic version of the BaumConnes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the BaumConnes conjecture in terms of twisted WeylMoyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the BaumConnes Conjecture actually suggests a strategy for quantizing such singular spaces.