We develop an integrated framework for studying income convergence that incorporates traditional measures of beta-convergence and sigma-convergence. These concepts are formally linked by a measure of re-ranking (or leapfrogging). Our proposed measure of beta-convergence allows for nonlinearities in the growth process and explicitly identifies the contribution of faster growth among low-income regions to reductions in overall inequality. To develop our framework we exploit the close links that exist between studies of income convergence and studies that examine the progressivity of the tax system. We illustrate our approach by examining both cross-country and regional income dynamics.