It is a classical and well-established result that a unirational algebraic variety of dimension ≪ 2 is necessarily rational. It is also generally agreed, though not so well established, that this result is no longer true for varieties of dimension three. In this connexion, the critical example has been the Enriques threefold E, asserted by Roth (1) to be unirational but not rational. The purpose of this note is to point out a fallacy in the proof of the non-rationality of E and, incidentally, to resolve a difficulty raised by Serre (2) in which E would appear to feature as a counter-example to a well-established general theorem. As far as the author is aware, there are no other examples of nonrational unirational threefolds, so that the question of their existence is still open.