  The mergers of binaries containing neutron stars and stellar-mass black holes
are the most promising sources for direct detection in gravitational waves by
the interferometers Advanced LIGO and Virgo over the next few years. The
concurrent detection of electromagnetic emission from these events would
greatly enhance the scientific return of these discoveries. Here we review the
state of the art in modeling the electromagnetic signal of neutron star binary
mergers across different phases of the merger and multiple wavelengths. We
focus on those observables which provide the most sensitive diagnostics of the
merger physics and the contribution to the synthesis of rapid neutron capture
($r$-process) elements in the Galaxy. We also outline expected future
developments on the observational and theoretical sides of this rapidly
evolving field.
