Mixed waste-solvent streams generated in pharmaceutical and specialty chemical manufacturing typically form multicomponent azeotropic mixtures. This highly nonideal behavior often prevents separation and recovery of the solvents. Batch distillation remains an important separation technology for these industries. The sequence of pure component and azeotropic cuts produced in batch distillation depends strongly on the feed composition. For ternary- and four-component mixtures the separation behavior can be studied graphically in residue-curve maps. For mixtures with more components, this approach is infeasible. Other tools are therefore required. Deficiencies in earlier work on ternary systems are demonstrated and addressed as well as a complete set of concepts to describe batch distillation of an azeotropic mixture with an arbitrary number of components. The body of theory is derived from the fields of nonlinear dynamics and topology.