We report on a novel compact laser-driven neutron source with an unprecedented short pulse duration (<50 ps) and high peak flux (>10(18) n/cm(2)/s), an order of magnitude higher than any existing source. In our experiments, high-energy electron jets are generated from thin (<3 μm) plastic targets irradiated by a petawatt laser. These intense electron beams are employed to generate neutrons from a metal converter. Our method opens venues for enhancing neutron radiography contrast and for creating astrophysical conditions of heavy element synthesis in the laboratory.