In this work we develop a general method to examine cooperative diffusion, i.e., the partial dynamic structure factors, in multicomponent colloidal mixtures of spherical particles. Using a multivariable projection operator formalism based on the many-body Smoluchowski diffusion equation, we derive an exact microscopic expression for the matrix of irreducible memory functions associated with the partial dynamic structure factors and the long-time cooperative diffusion coefficients of the system. Starting from this microscopic expression, we present a derivation of a self-consistent mode coupling scheme (MCS) for cooperative diffusion in colloidal mixtures. This scheme accounts not only for the direct particle interactions, but also for the far-field part of the solvent mediated hydrodynamic interactions. Combined with our recent work on the mode coupling theory of tracer-diffusion and linear viscoelasticity of colloidal suspensions, this MCS provides a unified method for calculating dynamic and low-shear rheological properties of multicomponent colloidal dispersions. The MCS can be used to study polydispersity and mixing effects in concentrated colloidal systems. Some applications related to cooperative diffusion, interdiffusion, and glass transition in binary mixtures are discussed to illustrate the method.