Over the past several years, a number of conjectures related to negative dependence of Bernoulli random variables $X_i$ have been made by R. Pemantle, D. Wagner and others. Among them are: (a) The Rayleigh property (which is a property of the generating polynomial of the $X_i$'s) implies the ultra logconcavity (ULC) of the rank sequence $a_k=P (\sum_iX_i=k)$. (b) The Rayleigh property implies negative association (which is a type of negative dependence). (c) The symmetric exclusion process (which is one of the main models in the field of interacting particle systems) with product initial distribution is negatively associated at positive times. These are motivated by problems in probability, mathematical physics, and combinatorics. In the latter context, the connection is with Mason's conjecture for certain classes of matroids. We will discuss these and other conjectures. Among the results: (a) is false and (c) is true, while (b) remains open. Furthermore, a stronger form of the Rayleigh property does imply both ULC and negative association. As a consequence of (c), we obtain distributional limit theorems for certain functionals of the symmetric exclusion process. A key tool in the positive results is a property of polynomials known as stability. Much of this is joint work with J. Borcea and P. Br\"and\'en.