We have placed limits on the cosmological significance of gas-rich low surface-brightness ( LSB ) galaxies as a proportion of the total population of gas-rich galaxies by carrying out a very deep survey ( HIDEEP ; Minchin et al . 2003 ) for neutral hydrogen ( H i ) with the Parkes multibeam system . Such a survey avoids the surface-brightness selection effects that limit the usefulness of optical surveys for finding LSB galaxies . To complement the HIDEEP survey we have digitally stacked eight 1-hour R -band Tech Pan films from the UK Schmidt Telescope covering 36 square degrees of the survey area to reach a very deep isophotal limit of 26.5 R mag arcsec ^ { -2 } . At this level , we find that all of the 129 H i sources within this area have optical counterparts and that 107 of them can be identified with individual galaxies . We have used the properties of the galaxies identified as the optical counterparts of the H i sources to estimate the significance of LSB galaxies ( defined to be those at least 1.5 magnitudes dimmer in effective surface-brightness than the peak in the observed distribution seen in optical surveys ) . Two different methods of correcting for ease-of-detection do not yield significantly different results : LSB galaxies make up 62 \pm 37 per cent of gas-rich galaxies by number according to our first method ( weighting by H i mass function ) , which includes a correction for large scale structure , or 51 \pm 20 per cent when calculated by our second method ( 1 / V _ { max } correction ) . We also find that LSB galaxies provide 30 \pm 10 per cent of the contribution of gas-rich galaxies to the neutral hydrogen density of the Universe , 7 \pm 3 per cent of their contribution to the luminosity density of the Universe ; 9 \pm 4 of their contribution to the baryonic mass density of the Universe , 20 \pm 10 per cent of their contribution to the dynamical mass density of the Universe and 40 \pm 20 per cent of their cross-sectional area . We do not find any ‘ Crouching Giant ’ LSB galaxies such as Malin 1 , nor do we find a population of extremely low surface-brightness galaxies not previously found by optical surveys . Such objects must either be rare , gas-poor or outside the survey detection limits .