It is well known that when a flexible or flexibly-mounted structure is placed perpendicular to the flow of a Newtonian fluid, it can oscillate due to the shedding of separated vortices at high Reynolds numbers. If the same flexible object is placed in non-Newtonian flows, however, the structure's response is still unknown. The main objective of this thesis is to introduce a new field of viscoelastic fluid-structure interactions by showing that the elastic instabilities that occur in the flow of viscoelastic fluids can drive the motion of a flexible structure placed in its path. Unlike Newtonian fluids, the flow of viscoelastic fluids can become unstable at infinitesimal Reynolds numbers due to the onset of a purely elastic flow instability. This instability occurs in the absence of nonlinear effects of fluid inertia and the Reynolds number of the flows studied here are in the order of 10-4. When such an elastic flow instability occurs in the vicinity of a flexible structure, the fluctuating fluid forces exerted on the structure grow large enough to cause a structural instability which in turn feeds back into the fluid resulting in a flow instability. Nonlinear periodic oscillations of the flexible structure are observed which have been found to be coupled to the time-dependent growth and decay of viscoelastic stresses in the wake of the structure. Presented in this thesis are the results of an investigation of the interaction occurring in the flow of a viscoelastic wormlike micelle solution past a flexible rectangular sheet. The structural geometries studied include: flexible sheet inclinations at 20°, 45° and 90° and flexible sheet widths of 5mm and 2.5mm. By varying the flow velocity, the response of the flexible sheet has been characterized in terms of amplitude and frequency of oscillations. Steady and dynamic shear rheology and filament stretching extensional rheology measurements are conducted in order to characterize the viscoelastic wormlike micelle solution. Bright field images show the deformation of the flexible sheet during an unstable oscillation while flow-induced birefringence images highlight the viscoleastic fluid stresses produced in the wake of the flexible sheet.