Keywords: Elasticity, viscoelastic instability, nonlinear transitions, drag-reducing polymers Abstract: Taylor-Couette flow (i.e., flow between concentric, rotating cylinders) has long served as a paradigm for studies of hydrodynamic stability. For Newtonian fluids, the rich cascade of transitions from laminar, Couette flow to turbulent flow occurs through a set of well-characterized flow states that depend on the Reynolds numbers of both the inner and outer cylinders (Rei and Reo). While extensive work has been done on (a) the effects of weak viscoelasticity on the first few transitions for Reo = 0 and (b) the effects of strong viscoelasticity in the limit of vanishing inertia (Rei and Reo both vanishing), the viscoelastic Taylor-Couette problem presents an enormous parameter space, much of which remains completely unexplored. Here we describe our recent experimental efforts to examine the effects of drag reducing polymers on the complete range of flow states observed in the Taylor-Couette problem. Of particular importance in the present work is 1) the rheological characterization of the test solutions via both shear and extensional (CaBER) rheometry, 2) the wide range of parameters examined, including Rei, Reo, and Elasticity number El, and 3) the use of a consistent, conservative protocol for accessing flow states. We hope to gain insights into the roles of weak elasticity and of co- and counter-rotation on nonlinear transitions in this flow.