Viscoelastic instabilities are of fundamental importance to understanding the physics of complex fluids and of practical importance to materials processing and fluid characterization. Significant progress has been made over the past 15 years in understanding instabilities in viscoelastic flows with curved streamlines and is reviewed here. Taylor-Couette flow, torsional flow between a cone and plate, and torsional flow between parallel plates have received special attention due to both the basic significance of these flows and their critical role in rheometry. First, we review the criteria for determining when these flows become unstable due to elasticity in the absence of inertia, and discuss the generalization of these criteria to more complex flows with curved streamlines. Then, focusing on experiments and simulations in the Taylor-Couette problem, we review how thermal sensitivity (i.e., the dependence of fluid viscosity and elasticity on temperature) and inertia affect the stability of viscoelastic flows. Finally, we conclude with some general thoughts on unresolved issues and remaining challenges related to viscoelastic instabilities.