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The penetration of a helical magnetic perturbation into a rotating tokamak plasma is investigated. In the linear regime, it is found that unless the frequency of the imposed perturbation matches closely to one of the natural mode frequencies, reconnection at the rational surface is suppressed by a large factor. In order to deal with the problem in the nonlinear regime a theory of propagating, constant-ψ magnetic islands is developed. This theory is valid provided the island width greatly exceeds any microscopic scale length (but still remains small compared with the minor radius), and the magnetic Reynolds number of the plasma is sufficiently large. An island width evolution equation is obtained which, in addition to the usual Rutherford term, contains a stabilizing term due ultimately to the inertia of the plasma flow pattern set up around the propagating island. A complete solution is presented for the case where the island and its associated flow pattern are steady. In the nonlinear regime, a fairly sharp threshold is predicted for the magnitude of the applied perturbation. Below this threshold, the induced islands are rotationally suppressed and partially dragged along by the rotating plasma, and above it the islands are virtually fully reconnected and ‘‘locked’’ at the applied frequency of the perturbation. Numerical results from an initial value code are presented, which show good agreement with the analytic predictions. Finally, it is demonstrated that these theories can be used to interpret data recently obtained from the COMPASS-C device [Controlled Fusion and Plasma Heating 1990 (EPS, Geneva, 1990), Vol. 1, p. 379]. In particular, a positive explanation is given of why in some cases an applied quasistatic resonant magnetic perturbation can stabilize magnetohydrodynamic modes, but in others leads to a disruption.