  We present a catalog of hard X-ray serendipitous sources detected in the
first 80 months of observations by the Nuclear Spectroscopic Telescope Array
(NuSTAR). The NuSTAR serendipitous survey 80-month (NSS80) catalog has an
unprecedented $\sim$ 62 Ms of effective exposure time over 894 unique fields (a
factor of three increase over the 40-month catalog), with an areal coverage of
$\sim $36 deg$^2$, larger than all NuSTAR extragalactic surveys. NSS80 provides
1274 hard X-ray sources in the $3-24$ keV band (822 new detections compared to
the previous 40-month catalog). Approximately 76% of the NuSTAR sources have
lower-energy ($<10$ keV) X-ray counterparts from Chandra, XMM-Newton, and
Swift-XRT. We have undertaken an extensive campaign of ground-based
spectroscopic follow-up to obtain new source redshifts and classifications for
427 sources. Combining these with existing archival spectroscopy provides
redshifts for 550 NSS80 sources, of which 547 are classified. The sample is
primarily composed of active galactic nuclei (AGN), detected over a large range
in redshift ($z$ = 0.012-3.43), but also includes 58 spectroscopically
confirmed Galactic sources. In addition, five AGN/galaxy pairs, one dual AGN
system, one BL Lac candidate, and a hotspot of 4C 74.26 (radio quasar) have
been identified. The median rest-frame $10-40$ keV luminosity and redshift of
the NSS80 are $\langle{L_\mathrm{10-40 keV}}\rangle$ = 1.2 $\times$ 10$^{44}$
erg s$^{-1}$ and $\langle z \rangle = 0.56$. We investigate the optical
properties and construct composite optical spectra to search for subtle
signatures not present in the individual spectra, finding an excess of redder
BL AGN compared to optical quasar surveys predominantly due to the presence of
the host-galaxy and, at least in part, due to dust obscuration.
