It has been four decades since Apollo 16 returned the first wide-field UV imagery of the Earth and revealed the vast extent of exospheric hydrogen (H) atoms around the planet. Since that time, appreciation has grown regarding the significance of this outermost atmospheric layer, whose charge exchange interaction with ambient ions dissipates magnetospheric energy, generates the energetic neutral atoms (ENAs) widely used for remote sensing of the ring current dynamics during geomagnetic storms, and accelerates gravitational escape and thus permanent atmospheric evolution. Despite the importance of Earth’s H exosphere to the solar-terrestrial system, however, current understanding of its global structure and dynamical evolution is insufficient, such that the origin of persistent discrepancies between measurements and models remains unresolved. Remote sensing of UV emission from geocoronal H atoms, generated through resonant scattering of solar radiation at 121.6 nm (Lyman-alpha) is the only empirical means available to investigate the terrestrial exosphere. In this work, we present robust tomographic-based techniques we have developed in recent years to estimate the 3-D, global and time-dependent H-density distributions during quiet and storm-time from observations of its optically thin emission at Lyman-α acquired from the Lyman-alpha detectors onboard the NASA TWINS satellites. Several examples of recent 2D and 3D data analyses will be used to demonstrate the current state-of-the-art, reveal surprising exospheric phenomenon, and motivate future work.