The CLEAN algorithm is widely used in the deconvolution of radio astronomical images. This paper reviews the theoretical basis of the method and shows that the progress of the algorithm may be analysed in terms of a transient phase followed by an asymptotic phase. The convergence result by Schwarz refers to the asymptotic phase, and this result is extended to show how and why the algorithm can diverge under certain conditions. Although these conditions frequently hold in practice, the choice of loop gain and the stopping criterion are such that the algorithm is rarely allowed to continue to the stage where these considerations are important. An analysis of the transient phase shows that the corrugation effect frequently observed when attempting to CLEAN sources with extended emission begins in this phase but that its evolution depends on the same criteria as those which affect the asymptotic phase. The Smoothness Stabilized CLEAN (SSC) algorithm when used with a small loop gain is effective in reducing the corrugation effect, but at the cost of affecting the asymptotic behaviour of the algorithm. It is shown theoretically and by a one-dimensional example that the unique solution to which this algorithm converges does not have the desirable properties of sidelobe suppression usually attributed to it. By examining the SSC iterations in this case, we see that what is usually regarded as the solution of the SSC algorithm is in fact an early iterate of a process which ultimately converges to the undesirable solution.