We report the discovery of two transiting Neptunes by the HATSouth survey . The planet HATS-37Ab has a mass of 0.099 \pm 0.042 M _ { J } ( 31.5 \pm 13.4 M _ { \oplus } ) and a radius of 0.606 \pm 0.016 R _ { J } , and is on a P = 4.3315 days orbit around a V = 12.266 \pm 0.030 mag , 0.843 _ { -0.012 } ^ { +0.017 } M _ { \sun } star with a radius of 0.877 _ { -0.012 } ^ { +0.019 } R _ { \sun } . We also present evidence that the star HATS-37A has an unresolved stellar companion HATS-37B , with a photometrically estimated mass of 0.654 \pm 0.033 M _ { \sun } . The planet HATS-38b has a mass of 0.074 \pm 0.011 M _ { J } ( 23.5 \pm 3.5 M _ { \oplus } ) and a radius of 0.614 \pm 0.017 R _ { J } , and is on a P = 4.3750 days orbit around a V = 12.411 \pm 0.030 mag , 0.890 _ { -0.012 } ^ { +0.016 } M _ { \sun } star with a radius of 1.105 \pm 0.016 R _ { \sun } . Both systems appear to be old , with isochrone-based ages of 11.46 _ { -1.45 } ^ { +0.79 } Gyr , and 11.89 \pm 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 .