  We present measurements of the evolution of normal-galaxy X-ray emission from
$z \approx$ 0-7 using local galaxies and galaxy samples in the 6 Ms Chandra
Deep Field-South (CDF-S) survey. The majority of the CDF-S galaxies are
observed at rest-frame energies above 2 keV, where the emission is expected to
be dominated by X-ray binary (XRB) populations; however, hot gas is expected to
provide small contributions to the observed- frame < 1 keV emission at $z < 1$.
We show that a single scaling relation between X-ray luminosity ($L_{\rm X}$)
and star-formation rate (SFR) is insufficient for characterizing the average
X-ray emission at all redshifts. We establish that scaling relations involving
not only SFR, but also stellar mass ($M_\star$) and redshift, provide
significantly improved characterizations of the average X-ray emission from
normal galaxy populations at $z \approx$ 0-7. We further provide the first
empirical constraints on the redshift evolution of X-ray emission from both
low-mass XRB (LMXB) and high-mass XRB (HMXB) populations and their scalings
with $M_\star$ and SFR, respectively. We find $L_{\rm 2-10 keV}({\rm
LMXB})/M_\star \propto (1 + z)^{2-3}$ and $L_{\rm 2-10 keV}({\rm HMXB})$/SFR
$\propto (1 + z)$, and show that these relations are consistent with XRB
population-synthesis model predictions, which attribute the increase in LMXB
and HMXB scaling relations with redshift as being due to declining host galaxy
stellar ages and metallicities, respectively. We discuss how emission from XRBs
could provide an important source of heating to the intergalactic medium in the
early Universe, exceeding that of active galactic nuclei.
