Chandra is detecting a significant population of normal and starburst galaxies in extremely deep X-ray exposures. For example, approximately 15% of the sources arising in the 2 Ms Chandra Deep Field-North survey are fairly normal galaxies, where normal means Milky Way-type X-ray emission rather than simply exhibiting an optically normal spectrum. Many of these galaxies are being detected at large look-back times z=0.1-0.5, allowing the study of the evolution of X-ray binary populations over significant cosmological timescales. We are also detecting individual off-nuclear ultraluminous X-ray sources (e.g., X-ray binaries), providing the first direct constraints on the prevalence of lower-mass black holes at significantly earlier times. The X-ray emission from such normal galaxies may also be a useful star-formation rate indicator, based on radio/X-ray cross-identifications. We describe the contribution of normal galaxies to the populations which make up the X-ray background and present their directly measured X-ray number counts. We find that normal and starburst galaxies should dominate the 0.5--2 keV number counts at X-ray fluxes fainter than 7e-18 erg cm^{-2} s^{-1} (thus they will outnumber the mighty AGN). Finally, we look to the future, suggesting that it is important that the population of X-ray faint normal and starburst galaxies be well constrained in order to design the next generation of X-ray observatories.
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