The first detection of a binary neutron star merger through gravitational waves and photons marked the dawn of multi-messenger astronomy with gravitational waves, and it greatly increased our insight in different fields of astrophysics and fundamental physics. However, many open questions on the physical process involved in a compact binary merger still remain and many of these processes concern plasma physics. With the second generation of gravitational wave interferometers approaching their design sensitivity, the new generation under design study, and new X-ray detectors under development, the high energy Universe will become more and more a unique laboratory for our understanding of plasma in extreme conditions. In this review, we discuss the main electromagnetic signals expected to follow the merger of two compact objects highlighting the main physical processes involved and some of the most important open problems in the field.