We investigate the dynamics and evolution of coalescing neutron stars. The three-dimensional Newtonian equations of hydrodynamics are integrated by the `Piecewise Parabolic Method' However, we do include the effects of the emission of gravitational waves on the hydrodynamics. The properties of neutron star matter are described by the equation of state of Lattimer & Swesty. In addition to the fundamental hydrodynamic quantities, density, momentum, and energy, we follow the time evolution of the electron density in the stellar gas. Energy losses and changes of the electron abundance due to the emission of neutrinos are taken into account by an elaborate ``neutrino leakage scheme'', which employs a careful calculation of the lepton number and energy source terms of all neutrino types. The grid is Cartesian and equidistant with a resolution of 64**3 or 128**3, which allows us to calculate the self-gravity via fast Fourier transforms.

extended abstract contribution to the proceedings of the 17th Texas Symposium, 2 pages, uuencoded compressed postscript.