  Neutron stars (NSs) are extraordinary not only because they are the densest
form of matter in the visible Universe but also because they can generate
B-fields ten orders of magnitude larger than those currently constructed on
Earth. The combination of extreme gravity with the enormous electromagnetic
(EM) fields gives rise to spectacular phenomena like those observed on August
2017 with the merger of a binary neutron star (NSNS) system, an event that
generated a gravitational wave (GW) signal, a short $\gamma$-ray burst (sGRB),
and a kilonova. This event serves as the highlight so far of the era of
multimessenger astronomy. In this review, we present the current state of our
theoretical understanding of compact binary mergers containing NSs as gleaned
from the latest general relativistic magnetohydrodynamic simulations. Such
mergers can lead to events like the one on August 2017, GW170817, and its EM
counterparts, GRB 170817 and AT 2017gfo. In addition to exploring the GW
emission from binary black hole-neutron star and NSNS mergers, we also focus on
their counterpart EM signals. In particular, we are interested in identifying
the conditions under which a relativistic jet can be launched following these
mergers. Such a jet is an essential feature of most sGRB models and provides
the main conduit of energy from the central object to the outer radiation
regions. Jet properties, including their lifetimes and Poynting luminosities,
the effects of the initial B-field geometries and spins of the coalescing NSs,
as well as their governing equation of state, are discussed. Lastly, we present
our current understanding of how the Blandford-Znajek mechanism arises from
merger remnants as the trigger for launching jets, if, when and how a horizon
is necessary for this mechanism, and the possibility that it can turn on in
magnetized neutron ergostars, which contain ergoregions, but no horizons.
