  Two neutron stars merge somewhere in the Universe approximately every 10
seconds, creating violent explosions observable in gravitational waves and
across the electromagnetic spectrum. The transformative coincident
gravitational-wave and electromagnetic observations of the binary neutron star
merger GW170817 gave invaluable insights into these cataclysmic collisions,
probing bulk nuclear matter at supranuclear densities, the jet structure of
gamma-ray bursts, the speed of gravity, and the cosmological evolution of the
local Universe, among other things. Despite the wealth of information, it is
still unclear when the remnant of GW170817 collapsed to form a black hole.
Evidence from other short gamma-ray bursts indicates a large fraction of
mergers may form long-lived neutron stars. We review what is known
observationally and theoretically about binary neutron star post-merger
remnants. From a theoretical perspective, we review our understanding of the
evolution of short- and long-lived merger remnants, including fluid,
magnetic-field, and temperature evolution. These considerations impact
prospects of detection of gravitational waves from either short- or long-lived
neutron star remnants which potentially allows for new probes into the hot
nuclear equation of state in conditions inaccessible in terrestrial
experiments. We also review prospects for determining post-merger physics from
current and future electromagnetic observations, including kilonovae and
late-time x-ray and radio afterglow observations.
