  We characterize an all-sky catalog of ${\sim} 8,400$ $\delta$ Scuti variables
in ASAS-SN, which includes ${\sim} 3,300$ new discoveries. Using distances from
\textit{Gaia} DR2, we derive period-luminosity relationships (PLRs) for both
the fundamental mode and overtone pulsators in the $W_{JK}$, $V$, Gaia DR2 $G$,
$J$, $H$, $K_s$ and $W_1$ bands. We find that the overtone pulsators have a
dominant overtone mode, with many sources pulsating in the second overtone or
higher order modes. The fundamental mode pulsators have metallicity dependent
periods, with $\log_{10} (\rm P){\sim}-1.1$ for $\rm [Fe/H]<-0.3$ and
$\log_{10} (\rm P){\sim}-0.9$ for $\rm [Fe/H]>0$ which leads to a
period-dependant scale height. Stars with $\rm P>0.100\,d$ are predominantly
located close to the Galactic disk ($\rm |Z|<0.5\,kpc$). The median period at a
scale height of $\rm Z{\sim}0\, kpc$ also increases with the Galactocentric
radius $\rm R$, from $\log_{10} (\rm P){\sim}-0.94$ for sources with $\rm R>9\,
kpc$ to $\log_{10} (\rm P){\sim}-0.85$ for sources with $\rm R<7\, kpc$, which
is indicative of a radial metallicity gradient. To illustrate potential
applications of this all-sky catalog, we obtained 30 min cadence, image
subtraction TESS light curves for a sample of 10 fundamental mode and 10
overtone $\delta$ Scuti stars discovered by ASAS-SN. From this sample, we
identified two new $\delta$ Scuti eclipsing binaries, ASASSN-V
J071855.62$-$434247.3 and ASASSN-V J170344.20$-$615941.2 with short orbital
periods of $\rm P_{orb}=2.6096$ d and $\rm P_{orb}=2.5347$ d respectively.
