  We review the current status of compact object simulations that are based on
the Smooth Particle Hydrodynamics (SPH) method. The first main part of this
review is dedicated to SPH as a numerical method. We begin by discussing
relevant kernel approximation techniques and discuss the performance of
different kernel functions. Subsequently, we review a number of different SPH
formulations of Newtonian, special- and general relativistic ideal fluid
dynamics. We particularly point out recent developments that increase the
accuracy of SPH with respect to commonly used techniques. The second main part
of the review is dedicated to the application of SPH in compact object
simulations. We discuss encounters between two white dwarfs, between two
neutron stars and between a neutron star and a stellar-mass black hole. For
each type of system, the main focus is on the more common, gravitational
wave-driven binary mergers, but we also discuss dynamical collisions as they
occur in dense stellar systems such as cores of globular clusters.
