This paper describes a new approach to the synthesis of fixed-structure and fixed-order controllers. Such controllers are required in many practical applications. A broad class of fixed-structure controller synthesis problems can be reduced to the determination of a real controller parameter vector (or simply, a controller) K = (k(1), k(2), ..., k(iota)), so that a given set of real or complex polynomials of the form P (s, K) := P-o (s) + k(1)P(1) (s) + ... + k(iota)P(iota) (s) is Hurwitz. The stability of the closed-loop system requires a real characteristic polynomial to be Hurwitz, while several performance criteria can be satisfied by ensuring that a family of complex polynomials is Hurwitz. A novel feature of this paper is the exploitation of the interlacing property (IP) of Hurwitz polynomials to construct arbitrarily tight approximations of the set of,stabilizing controllers. This is done by systematically constructing sets of linear inequalities in K. The union of the feasible sets of linear inequalities provides an approximation of the set of all controllers K, which render P(s, K) Hurwitz. As the number of sets of linear inequalities increases and approaches infinity, we show that the union of the feasible sets approaches the set of all stabilizing controllers of the desired structure. The main tools that are used in the construction of the sets of linear inequalities are the Hermite-Biehler theorem, Descartes' rule of signs, and its generalization. We provide examples of the applicability of the proposed methodology to the synthesis of fixed-order stabilizing controllers.