Viscoelastic flow modeled by the Oldroyd-B equations will be discussed from an analytical and computational perspective. First I will present a local energy decay theorem which applies to a large class of hyperbolic systems including the Oldyoryd-B model. This decay theorem is used to prove that global smooth solutions exist for small initial data. While small solutions are global, the problem for large data is much more complicated. I will present recent computational work on the Oldroyd-B equations which indicates that the system develops singularities exponentially in time at hyperbolic stagnation points in the flow. The singularities arise in the stress field of the flow and the algebraic structure of these singularities depends critically on an important elasticity parameter, the Weissenberg number. A local approximation to the solution at the hyperbolic stagnation point is constructed and there is excellent agreement between the local solution and the simulations. In addition, past a critical Weissenberg number the flow pattern becomes quite sensitive to time periodic perturbations of the background forcing (or changes in initial data) and there is a transition from small scale local mixing around the stable and unstable manifolds to global mixing in the fluid.