Recent and ongoing research in the distillation of nonideal mixtures is reviewed focusing on advances in the methodologies for the synthesis, design, analysis and control of separation sequences involving homogeneous and heterogeneous azeotropic towers. Maps of residue curves and distillation lines are examined, as well as geometric methods for the synthesis and design of separation sequences, trends in the steady-state and dynamic analysis of homogeneous and heterogeneous towers, the nonlinear behavior of these towers, and strategies for their control. Emphasis is placed on the methods of computing all of the azeotropes associated with a multicomponent mixture, on the features that distinguish azeotropic distillations from their zeotropic counterparts, on the potential for steady-state multiplicity, and on the existence of maximum and minimum reflux bounds. important considerations in the selection of entrainers are examined. For the synthesis of separation trains, when determining the feasible product compositions, the graphical methods are clarified, especially the conditions tinder which distillation boundaries can be crossed and bounding strategies under finite reflux. The application of geometric theory to locate the fixed points, at minimum reflux, is reviewed in connection with homotopy-continuation algorithms for this purpose. The use of homotopy-continuation algorithms, especially for the steady-state simulation of heterogeneous azeotropic distillations, is justified. Methods for phase stability analysis are reviewed in connection with the location of real bifurcation points at phase transitions, an important feature of algorithms for the dynamic simulation of heterogeneous azeotropic distillations.