Gravitational-wave astronomy is now a reality. During my time at Caltech, the Advanced LIGO and Virgo observatories have detected gravitational waves from dozens of compact binary coalescences. All of these gravitational-wave events occurred in the relatively local Universe. In the first part of this thesis, I will instead look towards the remote Universe, investigating what LIGO and Virgo may be able to learn about cosmologically-distant compact binaries via observation of the stochastic gravitational-wave background. The...