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We present the first full catalog and science results for the Nuclear Spectroscopic Telescope Array (NuSTAR) serendipitous survey. The catalog incorporates data taken during the first 40 months of NuSTAR operation, which provide approx. 20 Ms of effective exposure time over 331 fields, with an areal coverage of 13 deg2, and 497 sources detected in total over the 324 keV energy range. There are 276 sources with spectroscopic redshifts and classifications, largely resulting from our extensive campaign of ground-based spectroscopic follow-up. We characterize the overall sample in terms of the X-ray, optical, and infrared source properties. The sample is primarily composed of active galactic nuclei (AGNs), detected over a large range in redshift from z = 0.002 to 3.4 (median of [z] = 0.56), but also includes 16 spectroscopically confirmed Galactic sources. There is a large range in X-ray flux, from log(f_3-24 keV/erg/s/sq cm) approx. -14 to -11, and in rest-frame 10-40 keV luminosity, from log(L10-40 keV/erg/s) approx. 39 to 46, with a median of 44.1. Approximately 79% of the NuSTAR sources have lower-energy (<10 keV) X-ray counterparts from XMM-Newton, Chandra, and Swift XRT. The mid-infrared (MIR) analysis, using WISE all-sky survey data, shows that MIR AGN color selections miss a large fraction of the NuSTAR-selected AGN population, from approx. 15% at the highest luminosities (LX> 10(exp 44) erg/s) to 80 at the lowest luminosities (LX > 10(exp 43) erg/s).