The mergers of binaries containing neutron stars and stellar-mass black holes are among 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. 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 that 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.