  We introduce the class of {\em strongly Rayleigh} probability measures by
means of geometric properties of their generating polynomials that amount to
the stability of the latter. This class covers important models such as
determinantal measures (e.g. product measures, uniform random spanning tree
measures) and distributions for symmetric exclusion processes. We show that
strongly Rayleigh measures enjoy all virtues of negative dependence and we also
prove a series of conjectures due to Liggett, Pemantle, and Wagner,
respectively. Moreover, we extend Lyons' recent results on determinantal
measures and we construct counterexamples to several conjectures of Pemantle
and Wagner on negative dependence and ultra log-concave rank sequences.
