A polynomial with nonnegative coefficients is strongly log-concave if it and all of its derivatives are log-concave as functions on the positive orthant. This rich class of polynomials includes many interesting examples, such as homogeneous real stable polynomials and mixed volume polynomials of convex bodies. Its structure is intimately related to combinatorial objects called matroids and generalized permutahedra. This theory gives powerful tools for showing discrete log-concavity of finite sequences. Distributions coming from the coefficients of such polynomials can also be approximately sampled. In this talk I will go over the structural properties of this rich class of real polynomials and discuss applications in combinatorics and computer science.   This talk is based on joint work with Nima Anari, Kuikui Liu and Shayan Oveis Gharan.