We present the results of a survey of neutral hydrogen emission in the Small Magellanic Cloud (SMC) with the Australia Telescope Compact Array (ATCA). The survey consists of a mosaic of 320 separate pointings of the 375-m array, resulting in a resolution of 1.6arcmin (28pc, for a distance of 60kpc) over a field of 20deg^2^. The rms brightness temperature sensitivity is 1.4K, corresponding to an H I column density sensitivity of 4x^18^cm^-2^ for each velocity channel of width 1.6km/s. The HI distribution is complex and, on scales <=1kpc, appears to be dominated by the effects of expanding H I shells, which are probably driven by the combined effects of supernovae and stellar winds from massive stars. The picture of the SMC that arises from the current data seems to challenge the earlier belief that the SMC consists of two or more spatially separate structures with different systemic velocities. We find that the observed multiple components are, in many cases, caused by the combined effects of the numerous shells and supershells. Altogether, we identify six supershells (defined here as those with radii greater than 300pc) and 495 giant shells. For each of these, we measure positions. radii, velocities and expansion rates, and derive ages and kinetic energy requirements. The apparent age distribution of shells is remarkably narrow, with a mean age of 5.4Myr and an intrinsic dispersion of 2Myr. Southern shells appear to be older, on average, by 2.5Myr. The kinetic energy of the shells is a large fraction of the gravitational binding energy of the SMC, implying that further disintegration of the SMC will occur with time, and especially at the next close passage with the Large Magellanic Cloud (LMC) or the Galaxy, unless the SMC possesses a massive halo. Because of their interferometric nature, the images presented here are insensitive to structures of size >=0.6{deg}, and should not be used for deriving total H I column densities