Discoveries by LIGO, Virgo and KAGRA are informing our understanding of dense matter and stellar evolution. We have already used the gravitational-wave data of GW170817 - the first signal from merging neutron stars - to constrain the equation of state of dense matter in neutron stars. With more distant sources typically expected, the distribution of masses in compact binary mergers is also a key observable in the coming years of gravitational-wave astronomy; the heavy neutron-star merger GW190425 and the neutron-star/black-hole mergers GW200105 and GW20011 have already have revealed that our source population includes binary systems unlike those previously observed in our Galaxy. In this talk, I will discuss methods being developed to explore matter and mass properties for LIGO/Virgo neutron stars. I will discuss how current results fit with other neutron-star observations, outline prospects of learning about matter in the current Advanced-detector era, and extrapolate to the potential of next-generation gravitational-wave observatories like Cosmic Explorer to map new regions in the phase diagram of dense neutron-rich matter and explore the endpoints of stellar evolution.