As an introduction to the subject of future accelerators, it will be useful to consider briefly the main points of discussion at the three international conferences on a similar theme held in 1956, 1959 and 1961. In 1956 there were several laboratories, in different parts of the world, engaged in building machines based on the latest important new principle in accelerator design, namely, alternating-gradient focusing. There was a feeling, however, that the end of the road had not yet been reached, and at the 1956 Conference the success of earlier innovations encouraged the accelerator physicists to present a number of new ideas. Some of them were rather natural extensions of known principles, as, for example, a machine of fixed frequency with alternating-gradient focusing (F. F. A. G. ; see Kerst et al. 1956). This was also the first conference at which there were serious suggestions for colliding-beam experiments (Kerst 1956). The ideas presented by the Russian physicists were much more spectacular; in particular the suggestion of Budker (1956) for setting up very large neutralized electron currents to provide guiding fields in the mega-gauss region.