Geometric dynamical systems ideas have been very successful in determining traveling and standing waves in one space dimension.  Techniques that have proved important for their existence and stability include geometric singular perturbation theory, Lin’s Method, the Evans Function and the Maslov Index. Spatial Dynamics constitutes an approach to extending these ideas to higher-dimensional domains that have a quasi-one dimensional character, such as a channel. Through a  recent collaborative effort, an approach has been developed that gives a spatial dynamics formulation for waves and patterns on general domains. This approach will be explained and some of the challenges for making the theory truly effective will be discussed. This involves joint work with Margaret Beck (BU), Yuri Latushkin (Missouri and NYU), Alim Sukhtayev (Miami) and Graham Cox (Newfoundland).