  We present Symphony, a compilation of $262$ cosmological,
cold-dark-matter-only zoom-in simulations spanning four decades of host halo
mass, from $10^{11}$$-$$10^{15}~M_{\mathrm{\odot}}$. This compilation includes
three existing simulation suites at the cluster and Milky Way$-$mass scales,
and two new suites: $39$ Large Magellanic Cloud-mass
($10^{11}~M_{\mathrm{\odot}}$) and $49$ strong-lens-analog
($10^{13}~M_{\mathrm{\odot}}$) group-mass hosts. Across the entire host halo
mass range, the highest-resolution regions in these simulations are resolved
with a dark matter particle mass of $\approx 3\times 10^{-7}$ times the host
virial mass and a Plummer-equivalent gravitational softening length of $\approx
9\times 10^{-4}$ times the host virial radius, on average. We measure
correlations between subhalo abundance and host concentration, formation time,
and maximum subhalo mass, all of which peak at the Milky Way host halo mass
scale. Subhalo abundances are $\approx 50\%$ higher in clusters than in
lower-mass hosts at fixed sub-to-host halo mass ratios. Subhalo radial
distributions are approximately self-similar as a function of host mass and are
less concentrated than hosts' underlying dark matter distributions. We compare
our results to the semianalytic model $\mathrm{\texttt{Galacticus}}$, which
predicts subhalo mass functions with a higher normalization at the low-mass end
and radial distributions that are slightly more concentrated than Symphony. We
use $\mathrm{\texttt{UniverseMachine}}$ to model halo and subhalo star
formation histories in Symphony, and we demonstrate that these predictions
resolve the formation histories of the halos that host nearly all currently
observable satellite galaxies in the universe. To promote open use of Symphony,
data products are publicly available at
http://web.stanford.edu/group/gfc/symphony.
