We present results from sensitive, multi-epoch NuSTAR observations of the late-type star-forming galaxy M83 (d=4.6 Mpc), which is the first investigation to spatially resolve the hard (E>10 keV) X-ray emission of this galaxy. The nuclear region and ~ 20 off-nuclear point sources, including a previously discovered ultraluminous X-ray (ULX) source, are detected in our NuSTAR observations. The X-ray hardnesses and luminosities of the majority of the point sources are consistent with hard X-ray sources resolved in the starburst galaxy NGC 253. We infer that the hard X-ray emission is most likely dominated by intermediate accretion state black hole binaries and neutron star low-mass X-ray binaries (Z-sources). We construct the X-ray binary luminosity function (XLF) in the NuSTAR band for an extragalactic environment for the first time. The M83 XLF has a steeper XLF than the X-ray binary XLF in NGC 253, consistent with previous measurements by Chandra at softer X-ray energies. The NuSTAR integrated galaxy spectrum of M83 drops quickly above 10 keV, which is also seen in the starburst galaxies NGC253, NGC 3310 and NGC 3256. The NuSTAR observations constrain any AGN to be either highly obscured or to have an extremely low luminosity of $_{sim}^<$10$^{38}$ erg/s (10-30 keV), implying it is emitting at a very low Eddington ratio. An X-ray point source consistent with the location of the nuclear star cluster with an X-ray luminosity of a few times 10$^{38}$ erg/s may be a low-luminosity AGN but is more consistent with being an X-ray binary.
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