  In addition to their possible relevance to gamma-ray bursts, coalescing
binary neutron stars have long been recognized as important sources of
gravitational radiation that should become detectable with the new generation
of laser interferometers such as LIGO. Hydrodynamics plays an essential role
near the end of the coalescence when the two stars finally merge together into
a single object. The shape of the corresponding burst of gravitational waves
provides a direct probe into the interior structure of a neutron star and the
nuclear equation of state. The interpretation of the gravitational waveform
data will require detailed theoretical models of the complicated
three-dimensional hydrodynamic processes involved. Here we review the results
of our recent work on this problem, using both approximate quasi-analytic
methods and large-scale numerical hydrodynamics calculations on supercomputers.
We also discuss briefly the coalescence of white-dwarf binaries, which are also
associated with a variety of interesting astrophysical phenomena.
