  Univariate polynomials with only real roots -- while special -- do occur
often enough that their properties can lead to interesting conclusions in
diverse areas. Due mainly to the recent work of two young mathematicians,
Julius Borcea and Petter Br\"and\'en, a very successful multivariate
generalization of this method has been developed. The first part of this paper
surveys some of the main results of this theory of "multivariate stable"
polynomials -- the most central of these results is the characterization of
linear transformations preserving stability of polynomials. The second part
presents various applications of this theory in complex analysis, matrix
theory, probability and statistical mechanics, and combinatorics.
