Abstract. This paper summarises a two-hour discussion at the Ninth International Conference on Quasicrystals, including nearly 20 written comments sent afterwards, concerning (i) the meaning [if any] of clusters in quasicrystals; (ii) phason elasticity, and (iii) thermodynamic stabilisation of quasicrystals.This paper represents a two-hour discussion that we moderated at the Ninth International Conference on Quasicrystals. The discussion was divided into three parts, concerning (i) the meaning [if any] of clusters in quasicrystals; (ii) phason elasticity, and (iii) thermodynamic stabilisation of quasicrystals. To prime the participants, the discussion leaders prepared discussion papers in advance, which are printed separately [1, 2, 3].The colloquy was rather lively, considering that the topics are probably the most intractable questions in quasicrystals and the ones which have seen the least progress since 1985. There was somewhat less detailed discussion of specific experiments [done, or proposed] than might have been wished, but the discussion made absolutely clear that more experiments are needed in each of the three areas, if we are ever to resolve these questions.Included in this text are (i) notes from the oral comments taken at the discussion session; and (ii) written comments -about 18, almost evenly divided between theorists and experimentalists -which were sent afterwords by most of the participants who spoke during the session: