We present the first survey results of hard X-ray point sources in the Galactic Center (GC) region by NuSTAR. We have discovered 70 hard (3-79 keV) X-ray point sources in a 0.6 deg^2^ region around Sgr A* with a total exposure of 1.7 Ms, and 7 sources in the Sgr B2 field with 300 ks. We identify clear Chandra counterparts for 58 NuSTAR sources and assign candidate counterparts for the remaining 19. The NuSTAR survey reaches X-ray luminosities of ~4x and ~8x10^32^ erg/s at the GC (8 kpc) in the 3-10 and 10-40 keV bands, respectively. The source list includes three persistent luminous X-ray binaries (XBs) and the likely run-away pulsar called the Cannonball. New source-detection significance maps reveal a cluster of hard (>10 keV) X-ray sources near the Sgr A diffuse complex with no clear soft X-ray counterparts. The severe extinction observed in the Chandra spectra indicates that all the NuSTAR sources are in the central bulge or are of extragalactic origin. Spectral analysis of relatively bright NuSTAR sources suggests that magnetic cataclysmic variables constitute a large fraction (>40%-60%). Both spectral analysis and logN-logS distributions of the NuSTAR sources indicate that the X-ray spectra of the NuSTAR sources should have kT>20 keV on average for a single temperature thermal plasma model or an average photon index of {Gamma}=1.5-2 for a power-law model. These findings suggest that the GC X-ray source population may contain a larger fraction of XBs with high plasma temperatures than the field population.