  Mergers of binaries consisting of two neutron stars, or a black hole and a
neutron star, offer a unique opportunity to study a range of physical and
astrophysical processes using two different and almost orthogonal probes -
gravitational waves (GW) and electromagnetic (EM) emission. The GW signal
probes the binary and the physical processes that take place during the last
stages of the merger, while the EM emission provides clues to the material that
is thrown out following the merger. The accurate localization, which only the
EM emission can provide, also indicates the astrophysical setting in which the
merger took place. In addition, the combination of the two signals provides
constraints on the nature of gravity and on the expansion rate of the Universe.
The first detection of a binary neutron star merger by the LIGO-Virgo
collaboration, GW170817, initiated the era of multi-messenger GW-EM
astrophysics and demonstrated the great promise it holds. The event produced an
unprecedented data set, and although it was only a single event, it provided
remarkable results that revolutionized our knowledge of neutron star mergers.
GW170817 is especially exciting since we know that it is not one of a kind and
that many more events will be detected during the next decade. In this review,
I summarize, first, the theory of EM emission from compact binary mergers,
highlighting the unique information that the combined GW-EM detection provides.
I then describe the entire set of GW and EM observations of GW170817, and
summarize the range of insights that it offers. This includes clues about the
role that similar events play in the r-process elements budget of the Universe,
the neutron star equation of state, the properties of the relativistic outflow
that followed the merger, and the connection between neutron star binary
mergers and short gamma-ray bursts.
