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.
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