Abstract: In arbitrarily coupled dynamical systems (maps or ordinary differential equations), the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) and the formation of patterns from loss of stability of the synchronized states are two problems of current research interest. In this talk, three results are derived: First, general approaches are developed that yield constraints directly on the coupling strengths which ensure the stability of synchronized dynamics. Second, when the synchronized state becomes unstable generalized Turing patterns can be selectively realized by varying the coupling strengths. Third, given a desired spatiotemporal pattern, one is able to design coupling schemes which give rise to that pattern as the coupled system evolves.