The 'Local Volume HI Survey' (LVHIS) comprises deep HI spectral line and 20-cm radio continuum observations of 82 nearby, gas-rich galaxies, supplemented by multiwavelength images. Our sample consists of all galaxies with Local Group velocities vLG < 550 km s-1 or distances D < 10 Mpc that are detected in the HI Parkes All Sky Survey (HIPASS). Using full synthesis observations in at least three configurations of the Australia Telescope Compact Array (ATCA), we obtain detailed HI maps for a complete sample of gas-rich galaxies with δ ≃ -30°. Here we present a comprehensive LVHIS galaxy atlas, including the overall gas distribution, mean velocity field, velocity dispersion, and position-velocity diagrams, together with a homogeneous set of measured and derived galaxy properties. Our primary goal is to investigate the HI morphologies, kinematics, and environment at high resolution and sensitivity. LVHIS galaxies represent a wide range of morphologies and sizes; our measured HI masses range from~107 to 1010 M⊙, based on independent distance estimates. The LVHIS galaxy atlas (including FITS files) is available on-line.