The Faint Irregular Galaxies GMRT Survey (FIGGS) is a Giant Metrewave Radio Telescope (GMRT) based H i imaging survey of a systematically selected sample of extremely faint nearby dwarf irregular galaxies. The primary goal of FIGGS is to provide a comprehensive and statistically robust characterization of the neutral interstellar medium properties of faint, gas-rich dwarf galaxies. The FIGGS galaxies represent the extremely low mass end of the dwarf irregular galaxies population, with a median MB∼−13.0 and median H i mass of ∼3 × 107 M⊙, extending the baseline in mass and luminosity space for a comparative study of galaxy properties. The H i data are supplemented with observations at other wavelengths. In addition, distances accurate to ∼10 per cent are available for most of the sample galaxies. This paper gives an introduction to FIGGS, describes the GMRT observations and presents the first results from the H i observations. From the FIGGS data, we confirm the trend of increasing H i to optical diameter ratio with decreasing optical luminosity; the median ratio of DH i/DHo for the FIGGS sample is 2.4. Further, on comparing our data with aperture synthesis surveys of bright spirals, we find at best marginal evidence for a decrease in average surface density with decreasing H i mass. To a good approximation, the discs of gas-rich galaxies, ranging over three orders of magnitude in H i mass, can be described as being drawn from a family with fixed H i average surface density.