Hyperbolic polynomials are real polynomials with a simple reality condition on the zeros, reminiscent of characteristic polynomials of symmetric matrices. These polynomials appear in different areas of mathematics, including optimization, combinatorics and differential equations. We investigate the relation between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we realize as a linear slice of the cone of nonnegative polynomials. In this way we obtain information about determinantal representations and explicit sums-of-squares relaxations of hyperbolicity cones.