Compact objects observed in gravitational waves carry imprints of the physics of extremely dense matter.  Some signatures, such as accelerated phasing due to neutron star tidal deformability, are explicit and have already constrained the dense-matter equation of state. Others, especially matter observables associated with binaries containing black holes, have proven more elusive.  In this talk, I will point to promising avenues suggested by recent gravitational-wave observations, and discuss some first efforts in connecting extreme-matter physics to the observed population of binary black holes.