Viscoelastic forces are well-known to produce instabilities which are absent in Newtonian fluids and which may impose significant limitations on polymer processing operations such as extrusion or fiber spinning. Such instabilities are also present in the model problem of rotational shearing flow between concentric cylinders (Taylor-Couette flow). Our recent experimental work has focused on the Taylor-Couette problem both as a model for instabilities in polymer processing flows and as a test of constitutive equations for polymeric liquids. The isothermal Taylor-Couette system has long been a paradigm for studies of stability and transitions for Newtonian fluids; here, as the rotation rate of the inner cylinder is increased, the purely azimuthal base flow becomes unstable at a critical Reynolds number and is replaced by a stationary, axisymmetric vortex flow. This instability is driven by centrifugal forces and thus does not occur if the inner cylinder is held fixed and the outer cylinder is rotated. In the case of viscoelastic fluids, e.g. dilute solutions of polymers, a purely elastic instability occurs at vanishing Re. Here the flow is predicted to become unstable to an oscillatory, non-axisymmetric vortex flow at a critical value of fluid elasticity. Since the elastic instability is unrelated to centrifugal forces, it occurs for both rotation of the inner and rotation of the outer cylinder. While a purely elastic instability has been documented experimentally in a range of polymer solutions, and is indeed independent of which cylinder is rotated, there remain a number of discrepancies between experiments and predictions. Recent experiments have revealed that viscous heating in Newtonian fluids drives a transition to a new, oscillatory mode of instability at a critical Reynolds number which is significantly below that at which the isothermal, centrifugal transition is predicted and observed. A similar destabilization of viscoelastic modes through viscous heating may explain the discrepancies between the observed and predicted spatial and temporal symmetry of the disturbance flow and the critical conditions.