Abstract: We report the discovery of two transiting Neptunes by the HATSouth survey. The planet HATS-37Ab has a mass of 0.099 +/- 0.042 MJ (31.5.+/-.13.4M(circle dot)) and a radius of 0.606 +/- 0.016 R-J, and is on a P = 4.3315 day orbit around a V = 12.266 +/- 0.030 mag, 0.843(-0.012)(+0.017)M(circle dot) star with a radius of 0.877(-0.012)(+0.019) R-circle dot We also present evidence that the star HATS-37A has an unresolved stellar companion HATS-37B, with a photometrically estimated mass of 0.654 +/- 0.033.M-circle dot The planet HATS-38b has a mass of 0.074. 0.011MJ (23.5 +/- 3.5M(circle dot)) and a radius of 0.614 +/- 0.017 R-J, and is on a P = 4.3750 day orbit around a V = 12.411 +/- 0.030 mag, 0.890(-0.012)(+0.016) M-circle dot star with a radius of 1.105 +/- 0.016.R-circle dot Both systems appear to be old, with isochrone-based ages of 11.46(-1.45)(+0.79) Gyr, and 11.89 +/- 0.60 Gyr, respectively. Both HATS-37Ab and HATS-38b lie in the Neptune desert and are thus examples of a population with a low occurrence rate. They are also among the lowest-mass planets found from ground-based wide-field surveys to date.