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Neutron stars (NSs) are extraordinary not only because they are the densestform of matter in the visible Universe but also because they can generateB-fields ten orders of magnitude larger than those currently constructed onEarth. The combination of extreme gravity with the enormous electromagnetic(EM) fields gives rise to spectacular phenomena like those observed on August2017 with the merger of a binary neutron star (NSNS) system, an event thatgenerated 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 ofmultimessenger astronomy. In this review, we present the current state of ourtheoretical understanding of compact binary mergers containing NSs as gleanedfrom the latest general relativistic magnetohydrodynamic simulations. Suchmergers can lead to events like the one on August 2017, GW170817, and its EMcounterparts, GRB 170817 and AT 2017gfo. In addition to exploring the GWemission from binary black hole-neutron star and NSNS mergers, we also focus ontheir counterpart EM signals. In particular, we are interested in identifyingthe conditions under which a relativistic jet can be launched following thesemergers. Such a jet is an essential feature of most sGRB models and providesthe main conduit of energy from the central object to the outer radiationregions. 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 presentour current understanding of how the Blandford-Znajek mechanism arises frommerger remnants as the trigger for launching jets, if, when and how a horizonis necessary for this mechanism, and the possibility that it can turn on inmagnetized neutron ergostars, which contain ergoregions, but no horizons.

Multimessenger Binary Mergers Containing Neutron Stars: Gravitational Waves, Jets, and $\boldsymbol{\gamma}$-Ray Bursts