Dynamical systems theory offers some advantages over classical approaches to lagrangian transport in turbulent flows with organized structure, such as eddies and jets. The dynamical systems viewpoint allows us to see structure within a flow and to better understand what is causing the structure. As a result, the structure itself can be exploited rather than assumed to be non-existent or non-influential. For example, we have developed the means to apply dynamical systems theory to unsteady two-dimensional solenoidal velocity fields, and have analyzed a model of an oceanic jet, which is representative of the Gulf Stream. We have observed structure within the flow which has led us to a better understanding of how and why transport occurs across the jet. In addition, we have used the structure revealed by dynamical systems theory to compute the precise amount of flux across the jet and have observed the evolution of specific oceanic phenomena, such as "rings'', which are typically found near the Gulf Stream. The simplicity of these methods and their potential usefulness for studying transport will be emphasized.