Neutron star mergers emit strong gravitational-wave signals and
launch baryonic outflows, which generate observable signals in electromagnetic (EM) waves.
As demonstrated by the discovery of the neutron star merger GW170817, these multi-messanger signals
carry valuable information, e.g., the binary parameters and the nature of merger outflows. In the first part of my talk, I will focus on the inspiral and merger gravitational wave signals, from which we can extract the size of the neutron stars (neutron star EOS).
Here I will show numerical relativity simulations play an important role to compute waveforms with high precision.
In the second part of the talk, I will discuss physics of EM counterparts and what we learned from GW170817.
In particular, I will focus on (1) the kilonova in GW170817, pointing to neutron star mergers as the origin of r-process elements
and (2) the radio afterglow, showing a significant superluminal motion of a jet, which allows us to measure the jet's Lorentz factor and kinetic energy. The letter provides us with strong evidence that mergers are the progenitors of short gamma-ray bursts. Finally, I will discuss the Hubble constant measured from GW170817 and the precision of this measurement can be significantly improved by using the viewing angle information from the superluminal motion.