I'll report on progress on a program to understand global hypoellipticity of certain geometric (degenerate) Laplacians. Global hypoellipticity means that globally smooth data correspond to globally smooth solutions. This is a very weak property, which is compatible with propagation of singularities. Results from the 1990s about the d-bar Neumann problem (due to Boas, Straube, Christ etc.) hint to the existence of topological obstructions to global hypoellipticity. This talk is partially based on joint work with A. Martini.