Atmospheric gravity waves (GWs) are key drivers of the atmospheric circulation, but their representation in general circulation models (GCMs) is challenging, leading to significant biases in middle atmospheric circulations. Unresolved GW momentum transport in GCMs must be parameterized, but global directional GW observations are needed to constrain this. Here we present an 18-year climatology of directional stratospheric GW momentum flux (GWMF) from global AIRS/Aqua 3-D satellite observations during 2002 to 2019. Striking hemispheric asymmetries are found at high latitudes, including dramatic reductions and reversals of GWMF during sudden stratospheric warmings. During Southern Hemisphere winter, a lateral convergence of GWMF toward 60°S is found that has no Northern Hemisphere counterpart. In the tropics, we find that zonal GWMF in AIRS measurements is strongly modulated by the semiannual oscillation (SAO) but not the quasi-biennial oscillation (QBO). Our results provide guidance for future GW parameterizations needed to resolve long-standing biases in GCMs. Plain Language Summary Gravity waves (GWs) are traveling waves that can occur in geophysical fluid environments subject to the gravitational force. In the Earth's atmosphere, GWs transport momentum that helps to drive the atmospheric circulation, especially in the middle atmosphere. But for numerical weather and climate models, accurately simulating GWs has proven very challenging because of a lack of GW observations, leading to significant model biases. Here we present an 18-year climatology of GW observations near 40-km altitude for 2002 to 2019. For the first time, we present multidecadal global measurements of directional GW momentum flux derived from 3-D satellite observations from National Aeronautics and Space Administration's (NASA's) AIRS instrument. We find that during Southern Hemisphere winter, GWs travel laterally toward 60°S each year. This is significant because most models underestimate GW momentum near 60°S and do not typically include lateral propagation in their parameterizations. In the tropical stratosphere, we find no modulation of GWs in AIRS observations by the quasi-biennial oscillation (QBO). This is consistent with the hypothesis that GW-QBO interactions occur at shorter vertical wavelengths, which are invisible to AIRS. Our results will help to guide future GW parameterizations, leading to more accurate atmospheric simulations and ultimately better forecasts of weather and climate.