The growing catalog of gravitational-wave signals from compact-object mergers has allowed us to study the properties of black hole and neutron star binaries with increasing precision. However, the processes governing the formation and evolution of these systems and their electromagnetic counterparts remain largely unconstrained. The current observing run of the LIGO-Virgo-Kagra gravitational-wave detector network has already doubled the number of observed sources, and the next generation of detectors will be sensitive to nearly all compact-object mergers across cosmic time. In this talk, I will highlight how measurements of the population properties of black hole and neutron star binaries can be used to constrain the astrophysical processes driving their formation and evolution, especially when combined with multimessenger observations. I will conclude by describing novel methods to tackle the data-analysis challenges posed by next-generation gravitational-wave detectors to fully access the astrophysical and cosmological information within their reach.