In 1979, Helffer and Nourrigat made a very broad conjecture about the hypoellipticity of differential operators which are polynomials in a family of vectorfields. Their conjecture generalises a vast number of results—eg, the elliptic regularity theorem, Hörmander’s sums-of-squares theorem, and Rockland’s Theorem (proven by Hellfer-Nourrigat) on hypoellipticity for left invariant vector fields on graded nilpotent Lie groups. Helffer and Nourrigat proved several cases of the conjecture, but it has become newly accessible thanks to a startling observation by Debord and Skandalis which characterizes classical pseudodifferential operators in terms of Connes’ tangent groupoid. We will discuss how groupoidal methods can be vastly extended to resolve the Helffer-Nourrigat conjecture. This talk is based on joint work with E. Van Erp, with I. Androulidakis and O. Mohsen, and with N. Couchet.