Studies of the extragalactic Universe, from ultraviolet to infrared wavelengths, have been extremely effective at piecing together a basic picture of how stars in galaxies evolved throughout cosmic history.  At X-ray wavelengths, galaxy emission is dominated by hot gas and populations of X-ray binaries, the latter of which consist of black holes and neutron stars accreting material from normal stellar companions.  Hot gas in star-forming galaxies traces energetics from young and massive stars and X-ray binaries provide unique and important information regarding the star-formation histories and chemical evolution (metallicities) of their host galaxies.  These energetic phenomena have been proposed to play roles in the ionization of nebulae and long-range heating of the intergalactic medium in the early Universe. Furthermore, some X-ray binaries are expected to be predecessors and tracers of the gravitational-wave source populations that are now being detected by LIGO/VIRGO.  Using X-ray and multiwavelength observations (e.g., from Chandra, GALEX, Hubble, NuSTAR, Spitzer, Herschel, and other telescopes) of nearby and distant galaxies, as well as large-scale theoretical modeling, we are developing a framework detailing how X-ray binary populations and their host galaxies evolved together over the last 12 billion years (~90%) of cosmic history.  In this talk, I will describe some of the exciting new insights from our work, and I will highlight how new data sets, future observational facilities, and improved theoretical modeling will continue to improve our understanding of X-ray binaries, compact objects, and galaxies.