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A deuterium gas-puff load imploded by a multi-MA current driver from a large initial diameter could be a powerful source of fusion neutrons, a plasma neutron source (PNS). Unlike the beam-target neutrons produced in Z-pinch plasmas in the 1950s and deuterium-fiber experiments in the 1980s, the neutrons generated in deuterium gas-puffs with current levels achieved in recent experiments on the Z facility at Sandia National Laboratories could contain a substantial fraction of thermonuclear origin. For recent deuterium gas-puff shots on Z, our analytic estimates and one- and two-dimensional simulations predict thermal neutron yields ∼3×1013, in fair agreement with the yields recently measured on Z [C. A. Coverdale et al., Phys. Plasmas (to be published)]. It is demonstrated that the hypothesis of a beam-target origin of the observed fusion neutrons implies a very high Z-pinch-driver-to-fast-ions energy transfer efficiency, 5 to 10%, which would make a multi-MA deuterium Z-pinch the most efficient light-ion accelerator. No matter what mechanism is eventually determined to be responsible for generating fusion neutrons in deuterium gas-puff shots on Z, the deuterium neutron yield is shown to scale as Yn∼Im4, where Im is the peak current of the pinch. Theoretical estimates and numerical modeling of deuterium gas-puff implosions demonstrate that the yields of thermonuclear fusion neutrons that can be produced on ZR and the next-generation machines are sufficiently high to make PNS the most powerful, cost- and energy-efficient laboratory sources of 2.5−14MeV fusion neutrons, just like plasma radiation sources are the most powerful sources of soft and keV x rays. In particular, the predicted deuterium-tritium thermal neutron-producing capability of PNS driven by the next-generation ZR and ZX accelerators is ∼5×1016 and ∼1018, respectively.