Neutron stars host the densest stable matter in the universe. Accurately modeling their multi-messenger astrophysics relies on a detailed description of the equation of state above nuclear density, and astronomical observations ranging from x-ray timing to gravitational waves are in turn being used to constrain the properties of this dense matter.

The first gravitational-wave signal from a neutron star coalescence was observed in August 2017. I will describe how LIGO and Virgo have used this signal to constrain the equation of state of dense matter in neutron stars. The gravitational-wave transient was almost immediately associated with a short gamma-ray burst, and three-dimensional localization of the source using LIGO and Virgo data enabled a electromagnetic follow-up campaign that identified an associated kilonova in the galaxy NGC 4993. Additional information about dense matter emerges from such observations of light in concert with the gravitational waves.

A new observing run began in April 2019, and LIGO and Virgo have already sent public alerts for a couple of neutronstar merger candidates. I will outline prospects for learning about matter with gravitational waves in the current Advanced-detector era, and discuss how precision measurements with future ground-based observatories can map the phase diagram of dense neutron-rich matter.