The first observations to detect a population of distant galaxies directly in the submillimetre waveband have recently been made using the new Submillimetre Common-User Bolometer Array (SCUBA) on the James Clerk Maxwell Telescope (JCMT). The results indicate that a large number of distant galaxies are radiating strongly in this waveband. Here we discuss their significance for source confusion in future millimetre/submillimetre-wave observations of both distant galaxies and cosmic microwave background radiation (CMBR) anisotropies. Earlier estimates of such confusion involved significant extrapolation of the results of observations of galaxies at small redshifts; our new estimates do not, as they are derived from direct observations of distant galaxies in the submillimetre waveband. The results have important consequences for the design and operation of existing and proposed millimetre/submillimetre-wave telescopes: the Planck Surveyor mission will be confusion-limited at frequencies greater than 350 GHz, even in the absence of galactic dust emission; 1-sigma confusion noise of 0.44 mJy/beam is expected for the JCMT/SCUBA at a wavelength of 850 microns; and the sub-arcsecond resolution of large millimetre/submillimetre-wave interferometer arrays will be required in order to execute very deep galaxy surveys. For the latest information see astro-ph/9806063. ; Comment: 5 pages, 1 single-panel figure, 1 double-panel figure. MNRAS Pink Pages. This final published version includes updated references