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A compact high-flux, short-pulse neutron source would have applications from nuclear astrophysics to cancer therapy. Laser-driven neutron sources can achieve fluxes much higher than spallation and reactor neutron sources by reducing the volume and time in which the neutron-producing reactions occur by orders of magnitude. We report progress towards an efficient laser-driven neutron source in experiments with a cryogenic deuterium jet on the Texas Petawatt laser. Neutrons were produced both by laser-accelerated multi-MeV deuterons colliding with Be and mixed metallic catchers and by d ( d , n ) 3 He fusion reactions within the jet. We observed deuteron yields of 10 13 /shot in quasi-Maxwellian distributions carrying ∼ 8 − 10 % of the input laser energy. We obtained neutron yields greater than 10 10 /shot and found indications of a deuteron-deuteron fusion neutron source with high peak flux ( > 1 0 22 cm −2  s −1 ). The estimated fusion neutron yield in our experiment is one order of magnitude higher than any previous laser-induced dd fusion reaction. Though many technical challenges will have to be overcome to convert this proof-of-principle experiment into a consistent ultra-high flux neutron source, the neutron fluxes achieved here suggest laser-driven neutron sources can support laboratory study of the rapid neutron-capture process, which is otherwise thought to occur only in astrophysical sites such as core-collapse supernova, and binary neutron star mergers.