Recent advances in technology have encouraged proposals for new approaches to ATC in Europe and elsewhere. Two such proposals, both rather loosely framed, are for ‘free flight’ or for ‘seamless contracts’; otherwise ‘tubes of flight’. These concepts, and variations on them, aim to increase the traffic capacity of the airspace. They have been the subject of numerous published papers. Given the declared objects of these systems, it is surprising that, nearly without exception, these papers discuss the proposals only in qualitative terms. The present paper discusses idealised versions of these systems on the basis of quantitative studies. It is concluded that the ‘seamless contract’ is very probably unworkable in Europe. ‘free flight’ is workable in principle, but there is a need for investigation of possible mechanisms by which last-minute problems may be resolved.