This survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems. These are linear equations in a ring and result from a fractional representation of the systems involved. The cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant. One can then choose, in principle, the best controllers for various applications. These ideas evolved from early attempts to use polynomial equations in the design of discrete-time linear systems. By now they have been extended to continuous-time, infinite-dimensional, time-varying and non-linear systems.