We report the discovery of 10 transiting extrasolar planets by the HATSouth survey. The planets range in mass from the super-Neptune HATS-62b, with M_p_<0.179 M_J_, to the super-Jupiter HATS-66b, with M_p_=5.33 M_J_, and in size from the Saturn HATS-69b, with R_p_=0.94 R_J_, to the inflated Jupiter HATS-67b, with R_p_=1.69 R_J_. The planets have orbital periods between 1.6092 days (HATS-67b) and 7.8180 days (HATS-61b). The hosts are dwarf stars with masses ranging from 0.89 M_{sun}_ (HATS-69) to 1.56 M_{sun}_ (HATS-64) and have apparent magnitudes between V=12.276+/-0.020 mag (HATS-68) and V=14.095+/-0.030 mag (HATS-66). The super-Neptune HATS-62b is the least massive planet discovered to date with a radius larger than Jupiter. Based largely on the Gaia DR2 distances and broadband photometry, we identify three systems (HATS-62, HATS-64, and HATS-65) as having possible unresolved binary star companions. We discuss in detail our methods for incorporating the Gaia DR2 observations into our modeling of the system parameters and into our blend analysis procedures.