A population of distant dusty galaxies emitting in the submillimetre waveband has recently been detected using the Submillimetre Common-User Bolometer Array (SCUBA) camera on the James Clerk Maxwell Telescope (JCMT). This population can be used to trace the amount of high-redshift star formation activity that is obscured from view in the optical waveband by dust, and so is missing from existing inventories of star formation in the distant Universe. By including this population we can construct a complete and consistent picture of the history of star formation. The evolution of obscured star formation at redshifts less than unity is constrained by mid- and far-infrared counts of dusty galaxies. Activity increases with redshift z as (1 + z)(gamma) with gamma similar to 4, consistent with the form of evolution found in the optical waveband by the Canada-France Redshift Survey (CFRS) to z less than or similar to 1. The form of evolution at higher redshifts is constrained by both faint SCUBA counts and the intensity of background radiation in the millimetre/submillimetre waveband. We find that the total amount of energy emitted by dusty galaxies is about four times greater than that inferred from rest frame ultraviolet observations, and that a larger fraction of this energy is emitted at high redshifts. The simplest explanation for these results is that a large population of luminous, strongly obscured sources at redshifts z less than or similar to 5 is missing from optical surveys. We discuss the possible contribution of obscured active galactic nuclei to the submillimetre-wave background and counts. More accurate constraints on the history of star formation will be provided by determinations of the counts in several submillimetre wavebands and crucially by a reliable redshift distribution of the detected galaxies.