  The gravitational wave signals of black hole-neutron star (BHNS) binary
systems have now been detected, and future detections might be accompanied by
electromagnetic counterparts. BHNS mergers involve much of the same physics as
binary neutron star mergers: strong gravity, nuclear density matter, neutrino
radiation, and magnetic turbulence. They also share with binary neutron star
systems the potential for bright electromagnetic signals, especially gamma ray
bursts and kilonovae, and the potential to be significant sources of r-process
elements. However, BHNS binaries are more asymmetric, and their mergers produce
different amounts and arrangements of the various post-merger material
components (e.g. disk and dynamical ejecta), together with a more massive black
hole; these differences can have interesting consequences. In this chapter, we
review the modeling of BHNS mergers and post-merger evolution in numerical
relativistic hydrodynamics and magnetohydrodynamics. We attempt to give readers
a broad understanding of the answers to the following questions. What are the
main considerations that determine the merger outcome? What input physics must
(or should) go into a BHNS simulation? What have the most advanced simulations
to date learned?
