The growing catalog of gravitational-wave signals from compact object mergers has allowed us to study the properties of black holes and neutron stars more precisely than ever before and has opened a new window through which to probe the earliest moments in our universe’s history. In this talk, I will demonstrate how current and future gravitational-wave observations can be uniquely leveraged to learn about astrophysics and cosmology. With the current catalog of events detected by the LIGO and Virgo gravitational-wave detectors, I will present evidence for a correlation between the redshift and spin distributions of binary black holes and discuss its astrophysical implications. I will also present a data-driven estimate of the fraction of neutron star-black hole mergers that may be accompanied by an electromagnetic counterpart along with constraints on the neutron star equation of state using this population of sources. Finally, with the sensitivities expected for the next generation of gravitational-wave detectors, I will present the statistically optimal method for the simultaneous detection of a foreground of compact binary mergers and a stochastic gravitational-wave background from early-universe processes.