Our understanding of how to design distillation-based processes to separate mixtures displaying azeotropic behavior has grown enormously in the past quarter century. We elegantly sketch distillation column behavior on a composition diagram where we display VLE and column behavior using residue and distillation curves. In the presence of azeotropes, these curves partition composition space into distillation regions that trap the performance of individual columns. We can view liquid-liquid behavior as a tearing of composition space, a tear that always spawns from a minimum binary azeotrope. A material balance across a column section leads to a difference point, which we can use to understand column tray-by-tray and limiting behavior for ordinary, extractive and even reactive distillation. We demonstrate how to synthesize alternative separation processes, concentrating on those that produce pure products. We use boundary curvature, solvent addition, extractive agents, liquid-liquid behavior and strategically placed reaction to step across distillation boundaries. We show that these processes always contain recycles to gain feasibility but also to have economically attractive processes.