************************************************************************* 72054 ASCA made the first detection of the Geminga pulsar in the 2-5 keV band, resolving the nature of its hardest X-rays in favor of nonthermal emission, and discovering a complex new pulse shape at these energies. We now propose to make a longer observation to better characterize Geminga's X-ray spectrum and pulse profile in terms of their thermal and nonthermal components, and also to study Geminga in the 4-10 keV range. These data will bear upon a number of fundamental problems in neutron star physics including high-energy emission from magnetospheric accelerators, interior cooling models, the properties of magnetized atmospheres, polar cap heating by energetic particles, and long-term timing behavior. Several of these issues are discussed in the proposal. *************************************************************************