The behavior of Rossby waves on a shear flow in the presence of a nonlinear critical layer is studied, with particular emphasis on the role played by the critical layer in a Rossby wave resonance mechanism. Previous steady analyses are extended to the resonant case and it is found that the forced wave dominates the solution, provided the flow configuration is not resonant for the higher harmonics induced by the critical layer. Numerical simulations for the forced initial value problem show that the solution evolves towards the analysed steady state when conditions are resonant for the forced wave, and demonstrate some of the complications that arise when they are resonant for higher harmonics. In relating the initial value and steady problems, it is argued that the time dependent solution does not require the large mean flow distortion that Haberman (1972) found to be necessary outside the critical layer in the steady case.