We have conducted a submillimetre mapping survey of faint, gravitationally lensed sources, where we have targeted 12 galaxy clusters and additionally the New Technology Telescope (NTT) Deep Field. The total area surveyed is 71.5arcmin^2^ in the image plane; correcting for gravitational lensing, the total area surveyed is 40arcmin^2^ in the source plane for a typical source redshift z>>2.5. In the deepest maps, an image plane depth of 1{sigma} rms ~0.8mJy is reached. This survey is the largest survey to date to reach such depths. In total 59 sources were detected, including three multiply imaged sources. The gravitational lensing makes it possible to detect sources with flux density below the blank field confusion limit. The lensing-corrected fluxes range from 0.11 to 19mJy. After correcting for multiplicity, there are 10 sources with fluxes <2mJy of which seven have submJy fluxes, doubling the number of such sources known. Number counts are determined below the confusion limit. At 1mJy, the integrated number count is ~10^4^deg^-2^, and at 0.5mJy it is ~2x10^4^deg^-2^. Based on the number counts, at a source plan flux limit of 0.1mJy, essentially all of the 850-m background emission has been resolved. The dominant contribution (>50 per cent) to the integrated background arises from sources with fluxes S850 between 0.4 and 2.5mJy, while the bright sources S850>6mJy contribute only 10 per cent.