The science case and associated science requirements for a next-generation Very Large Array (ngVLA) are described, highlighting the five key science goals developed out of a community-driven vision of the highest scientific priorities in the next decade. Building on the superb cm observing conditions and existing infrastructure of the VLA site in the U.S. Southwest, the ngVLA is envisaged to be an interferometric array with more than 10 times the sensitivity and spatial resolution of the current VLA and ALMA, operating at frequencies spanning $sim1.2 - 116$,GHz with extended baselines reaching across North America. The ngVLA will be optimized for observations at wavelengths between the exquisite performance of ALMA at submm wavelengths, and the future SKA-1 at decimeter to meter wavelengths, thus lending itself to be highly complementary with these facilities. The ngVLA will be the only facility in the world that can tackle a broad range of outstanding scientific questions in modern astronomy by simultaneously delivering the capability to: (1) unveil the formation of Solar System analogues; (2) probe the initial conditions for planetary systems and life with astrochemistry; (3) characterize the assembly, structure, and evolution of galaxies from the first billion years to the present; (4) use pulsars in the Galactic center as fundamental tests of gravity; and (5) understand the formation and evolution of stellar and supermassive blackholes in the era of multi-messenger astronomy.
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