  We present the Aemulus $\nu$ simulations: a suite of 150 $(1.05 h^{-1}\rm
Gpc)^3$ $N$-body simulations with a mass resolution of $3.51\times 10^{10}
\frac{\Omega_{cb}}{0.3} ~ h^{-1} M_{\odot}$ in a $w\nu$CDM cosmological
parameter space. The simulations have been explicitly designed to span a broad
range in $\sigma_8$ to facilitate investigations of tension between large scale
structure and cosmic microwave background cosmological probes. Neutrinos are
treated as a second particle species to ensure accuracy to $0.5\, \rm eV$, the
maximum neutrino mass that we have simulated. By employing Zel'dovich control
variates, we increase the effective volume of our simulations by factors of
$10-10^5$ depending on the statistic in question. As a first application of
these simulations, we build new hybrid effective field theory and matter power
spectrum surrogate models, demonstrating that they achieve $\le 1\%$ accuracy
for $k\le 1\, h\,\rm Mpc^{-1}$ and $0\le z \le 3$, and $\le 2\%$ accuracy for
$k\le 4\, h\,\rm Mpc^{-1}$ for the matter power spectrum. We publicly release
the trained surrogate models, and estimates of the surrogate model errors in
the hope that they will be broadly applicable to a range of cosmological
analyses for many years to come.
