The conditions of multiphase equilibrium are solved for generic polydisperse systems. The case of multiple polydispersity is treated, where several properties (e.g., size, charge, shape) simultaneously vary from one particle to another. By developing a perturbative expansion in the width of the distribution of constituent species, it is possible to calculate the effects of polydispersity alone, avoiding difficulties associated with the underlying many-body problem. Explicit formulas are derived in detail, for the partitioning of species at coexistence and for the shift of phase boundaries due to polydispersity. Convective fractionation is quantified, whereby one property (e.g., charge) is partitioned between phases due to a driving force on another. To demonstrate the ease of use and versatility of the formulas, they are applied to models of a chemically polydisperse polymer blend, and of fluid-fluid coexistence in polydisperse colloid-polymer mixtures. In each case, the regime of coexistence is shown to be enlarged by polydispersity. (C) 2001 American Institute of Physics.