  We survey recent progress on the Greenfield-Wallach and Katok conjectures on
globally hypoelliptic and cohomology free vector fields and derive a proof of
the conjectures in dimension three. The argument is primarily based on recent
work of F. and J. Rodriguez Hertz which allows us to reduce the question to the
case of a Reeb flow for a contact form. The contact case is settled by invoking
the Weinstein conjecture (which has been recently announced by C. Taubes).
