We present extensions of the Gibbs-Duhem integration technique that permit its application to mixtures. The Gibbs-Duhem integration method provides a means to evaluate the equilibrium properties of coexisting phases efficiently and reliably by molecular simulation. In particular, the method prescribes how a sequence of simulations can be performed at state conditions that lie on the coexistence surface, thus it is particularly well suited for the mapping of complete phase diagrams. When applied to pure substances, the Clapeyron equation guides the choice of state conditions as the series proceeds. We present two extensions, based respectively on "semigrand" and "osmotic" forms of a generalized Clapeyron equation that we also present. Both methods require sampling of compositions during the simulation, but they differ in how this is accomplished. Simulations based on the semigrand method have steps in which molecules attempt to change their species identity, keeping the total number of molecules fixed; the osmotic simulations have moves in which insertions or deletions are attempted for one of the species, while the number of molecules of the other species remains always fixed. Each approach is seen to have certain advantages that depend on the nature of the mixture being studied. Both methods are demonstrated with application to three prototype binaries, and extension to multicomponent mixtures is discussed.