  We review the current status of studies of the coalescence of binary neutron
star systems. We begin with a discussion of the formation channels of merging
binaries and we discuss the most recent theoretical predictions for merger
rates. Next, we turn to the quasi-equilibrium formalisms that are used to study
binaries prior to the merger phase and to generate initial data for fully
dynamical simulations. The quasi-equilibrium approximation has played a key
role in developing our understanding of the physics of binary coalescence and,
in particular, of the orbital instability processes that can drive binaries to
merger at the end of their lifetimes. We then turn to the numerical techniques
used in dynamical simulations, including relativistic formalisms,
(magneto-)hydrodynamics, gravitational-wave extraction techniques, and nuclear
microphysics treatments. This is followed by a summary of the simulations
performed across the field to date, including the most recent results from both
fully relativistic and microphysically detailed simulations. Finally, we
discuss the likely directions for the field as we transition from the first to
the second generation of gravitational-wave interferometers and while
supercomputers reach the petascale frontier.
