Negative dependence models "repelling particles" in discrete probability theory and statistical mechanics. We introduce and study Lorentzian polynomials and Lorentzian distributions, which are relaxations of stable polynomials and strong Rayleigh distributions. We prove that Lorentzian distributions satisfy several negative dependence properties, and apply the results to concrete problems in matroid theory, linear algebra and statistical mechanics. This is joint work with June Huh.