The properties of matter are dominated by the electric interaction between charged particles. Neutrons, having no electric charge, serve as a unique tool for studies of material properties otherwise obscured by electric forces. Until recently the production of a high neutron flux was exclusive to reactor and accelerator-based facilities. The availability of tabletop particle sources based on high intensity lasers enables the realization of a high flux neutron source at a university-scale laboratory. I will report on a novel compact laser-driven neutron source with unprecedented short pulse duration (<50 ps) and high flux (>10¹⁸ neutrons/cm²/s), an order of magnitude higher than any existing source. Our results show a dramatic onset of high-energy electron jets generation from thin (< few microns) plastic targets irradiated by a petawatt laser. These intense electron beams were employed to generate neutrons from a metal converter. Our method opens venues for enhancing neutron radiography contrast, conducting time-resolved neutron-damage studies at their characteristic evolution time-scales and for creating astrophysical conditions of nucleosynthesis in the laboratory.