  The first detection of gravitational waves from a binary neutron star merger
(GW170817) and the accompanying electromagnetic emission has impressively
advanced our understanding of the merger process and has set some first
constraints on the macroscopic properties of neutron stars, with direct
implications for the high-density equation of state. We discuss work on neutron
star mergers focusing on the postmerger gravitational-wave emission. These
studies are based on numerical simulations of the merger and survey a large
sample of candidate equations of state for neutron star matter. The goal is to
connect observables with the underlying physics questions. This offers a way to
constrain the properties of high-density matter through the determination of
neutron star radii, as inferred by an empirical relation connecting the
dominant gravitational wave frequency peak in the postmerger phase to the
radius of nonrotating neutron stars of a certain mass. We clarify the physical
origin of secondary peaks and discuss a spectral classification scheme, based
on their relative strength. Observational prospects for the dominant and the
secondary peaks are also discussed. The threshold mass to black hole collapse
is connected by another empirical relation to the maximum mass and compactness
of nonrotating neutron stars, which can be derived semi-analytically. The
observation of GW170817 then sets an absolute minimum radius for neutron stars
of typical masses, based only on a minimal number of assumptions. We discuss
future prospects, in light of the planned upgrades of the current gravitational
wave detectors.
