This paper presents a simple, self-contained account of Gårding's theory of hyperbolic polynomials, together with a recent convexity result of Bauschke-Güler-Lewis-Sendov and an inequality of Gurvits. This account begins by establishing some new results. The first concerns the existence of a pointwise arrangement of the eigenvalues so that they become global real analytic functions. The second asserts that the associated “branches” are independent of the choice of hyperbolic direction. © 2013 Wiley Periodicals, Inc.