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) for neutral hydrogen (HI) 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/sq. arcsec. At this level, we find that all of the 129 HI 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 HI 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). We calculate the contribution of LSB galaxies to the total number, neutral hydrogen density, luminosity density, baryonic mass density, dynamical mass density and cross-sectional area of gas-rich galaxies. 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.
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