The effect of an insoluble surfactant on the stability of the core–annular flow of two immiscible fluids is investigated by a normal-mode linear analysis and by numerical simulations based on the immersed-interface method for axisymmetric perturbations. The results reveal that, although the Marangoni stress due to surfactant concentration variations is unable to initiate a new type of instability as in the case of two-dimensional two-layer channel flow, it does destabilize the interface by broadening the range of growing wavenumbers and by raising the growth rate of unstable perturbations. Numerical simulations for large-amplitude disturbances reveal that the surfactant plays an important role in determining the morphology of the interfacial structures developing in the nonlinear stages of the motion.