I shall describe work in progress with Bob Yuncken and Tyrone Crisp. The Bernstein transform is a map from one function space to another, like the Fourier transform or the Radon transform. It is part of Bernstein’s theory of smooth representations of p-adic groups, but the version I’m going to discuss comes from the tempered representation theory of real groups such as SL(2,R). The main problem in either context is to prove that it exists! I’ll explain what the Bernstein transform is designed to do, how to characterize it, at least for SL(2,R), using scattering theory for a wave equation (this involves a geometric space that is very similar to the tangent groupoid) and why I think that eventually it will be best understood from the point of view of C*-algebra theory (for the moment we rely on constructions that only work for smooth subalgebras).