  We review the dramatic progress in the simulations of compact objects and
compact-object binaries that has taken place in the first two decades of the
twenty-first century. This includes simulations of the inspirals and violent
mergers of binaries containing black holes and neutron stars, as well as
simulations of black-hole formation through failed supernovae and high-mass
neutron star--neutron star mergers. Modeling such events requires numerical
integration of the field equations of general relativity in three spatial
dimensions, coupled, in the case of neutron-star containing binaries, with
increasingly sophisticated treatment of fluids, electromagnetic fields, and
neutrino radiation. However, it was not until 2005 that accurate long-term
evolutions of binaries containing black holes were even possible. Since then,
there has been an explosion of new results and insights into the physics of
strongly-gravitating system. Particular emphasis has been placed on
understanding the gravitational wave and electromagnetic signatures from these
extreme events. And with the recent dramatic discoveries of gravitational waves
from merging black holes by the Laser Interferometric Gravitational Wave
Observatory and Virgo, and the subsequent discovery of both electromagnetic and
gravitational wave signals from a merging neutron star binary, numerical
relativity became an indispensable tool for the new field of multimessenger
astronomy.
