  We describe a wide class of polynomials, which is a natural generalization of
Hurwitz stable polynomials. We also give a detailed account of so-called
self-interlacing polynomials, which are dual to Hurwitz stable polynomials but
have only real and simple zeroes. All proofs are given using properties of
rational functions mapping the upper half-plane of the complex plane to the
lower half-plane. Matrices with self-interlacing spectra and other applications
of generalized Hurwitz polynomials are discussed.
