  We study the class of Lorentzian polynomials. The class contains homogeneous
stable polynomials as well as volume polynomials of convex bodies and
projective varieties. We prove that the Hessian of a nonzero Lorentzian
polynomial has exactly one positive eigenvalue at any point on the positive
orthant. This property can be seen as an analog of Hodge--Riemann relations for
Lorentzian polynomials. Lorentzian polynomials are intimately connected to
matroid theory and negative dependence properties. We show that matroids, and
more generally M-convex sets, are characterized by the Lorentzian property, and
develop a theory around Lorentzian polynomials. In particular, we provide a
large class of linear operators that preserve the Lorentzian property and prove
that Lorentzian measures enjoy several negative dependence properties. We also
prove that the class of tropicalized Lorentzian polynomials coincides with the
class of M-convex functions in the sense of discrete convex analysis. The
tropical connection is used to produce Lorentzian polynomials from M-convex
functions. We give two applications of the general theory. First, we prove that
the homogenized multivariate Tutte polynomial of a matroid is Lorentzian
whenever the parameter q satisfies $0 < q \le 1$. Consequences are proofs of
the strongest Mason's conjecture from 1972 and negative dependence properties
of the random cluster model model in statistical physics. Second, we prove that
the multivariate characteristic polynomial of an M-matrix is Lorentzian. This
refines a result of Holtz who proved that the coefficients of the
characteristic polynomial of an M-matrix form an ultra log-concave sequence.
