A nearly hundred-year old question by Fatou asks for a synthesis of the following two kinds of holomorphic dynamical systems under a common framework of holomorphic correspondences on the Riemann sphere: (a) Kleinian groups acting on the Riemann sphere (b) iteration of complex polynomials on the Riemann sphere. Sullivan's dictionary gave us a way of translating techniques from one of these fields to give results in the other. In a relatively recent development, building on Sullivan's dictionary, a bridge has been built between these two classes in the spirit of Bers' simultaneous uniformization theorem. New holomorphic dynamical systems on the Riemann sphere have thus been discovered that arise as combinations or matings of Kleinian groups and polynomials. In some cases, these single valued matings give rise to multi-valued algebraic correspondences on the Riemann sphere, partially fulfilling Fatou's dream. A particular consequence of these constructions is an analog of the compactness theorem for Bers slices of punctured sphere groups. In 1982, Thurston posed a number of questions that guided the development of the theory of Kleinian groups for the next 3 decades. With the above analog of Bers compactness in place, many of these questions reincarnate themselves in this new context. We will survey some of these developments and questions. This is joint work with Yusheng Luo and Sabyasachi Mukherjee.