An experimental and theoretical investigation of the stability of the viscoelastic flow of the Boger fluid between eccentric cylinders is considered. In our theoretical study, a local linear stability analysis for the flow of an Oldroyd-B fluid suggests that the flow is elastically unstable for all eccentricities. It is shown that the linear stability of eccentric cylinder flow, to leading order for small gap ratio, is very similar to that of Taylor-Dean flow (Y.L. Joo and E.S.G. Shaqfeh, Phys. Fluids A, 4(3) (1992) 524). Flow visualization experiments with a solution of high molecular polyisobutylene dissolved in a viscous solvent clearly show the transition from a purely azimuthal to a secondary torodial flow. Comparison of these experimental results with the local linear stability theory shows qualitative agreement with the predicted critical conditions for the onset of the non-inertial cellular instability when delta < 0.4, where delta is the eccentricity (made dimensionless with the average gap thickness). At higher eccentricities, experiment and theory cease to agree. Evidence is given that this disagreement is due to the non-local streamwise extension rate which is not included in the present theory, and whose magnitude increases with increasing delta. We look forward to a comprehensive investigation of this interesting phenomenon.