  The very first detection of gravitational waves from a neutron star binary
merger, GW170817, exceeded all expectations. The event was relatively nearby,
which may point to a relatively high merger rate. It was possible to extract
finite-size effects from the gravitational-wave signal, which constrains the
nuclear equation of state. Also, an electromagnetic counterpart was detected at
many wavebands from radio to gamma rays marking the begin of a new
multi-messenger era involving gravitational waves. We describe how
multi-messenger observations of GW170817 are employed to constrain the nuclear
equation of state. Combining the information from the optical emission and the
mass measurement through gravitational waves leads to a lower limit on neutron
star radii. According to this conservative analysis, which employs a minimum
set of assumptions, the radii of neutron stars with typical masses should be
larger than about 10.7~km. This implies a lower limit on the tidal
deformability of about 210, while much stronger lower bounds are not supported
by the data of GW170817. The multi-messenger interpretation of GW170817 rules
out very soft nuclear matter and complements the upper bounds on NS radii which
are derived from the measurement of finite-size effects during the pre-merger
phase. We highlight the future potential of multi-messenger observations and of
GW measurements of the postmerger phase for constraining the nuclear equation
of state. Finally, we propose an observing strategy to maximize the scientific
yield of future multi-messenger observations.
