The Newell-Littlewood numbers $N_{μ,ν,λ}$ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions $(μ,ν,λ)$ does $N_{μ,ν,λ}>0$ hold? The Littlewood-Richardson coefficient case is solved by the Horn inequalities (in work of A. Klyachko and A. Knutson-T. Tao). We extend these celebrated linear inequalities to a much larger family, suggesting a general solution.