  We solve Einstein's field equations coupled to relativistic hydrodynamics in
full 3+1 general relativity to evolve astrophysical systems characterized by
strong gravitational fields. We model rotating, collapsing and binary stars by
idealized polytropic equations of state, with neutron stars as the main
application. Our scheme is based on the BSSN formulation of the field
equations. We assume adiabatic flow, but allow for the formation of shocks. We
determine the appearance of black holes by means of an apparent horizon finder.
We introduce several new techniques for integrating the coupled
Einstein-hydrodynamics system. For example, we choose our fluid variables so
that they can be evolved without employing an artificial atmosphere. We also
demonstrate the utility of working in a rotating coordinate system for some
problems. We use rotating stars to experiment with several gauge choices for
the lapse function and shift vector, and find some choices to be superior to
others. We demonstrate the ability of our code to follow a rotating star that
collapses from large radius to a black hole. Finally, we exploit rotating
coordinates to evolve a corotating binary neutron star system in a
quasi-equilibrium circular orbit for more than two orbital periods.
