A vector Kirchhoff diffraction theory is used to obtain the Fraunhofer diffraction patterns and total power transmissions of circular apertures illuminated by partially coherent light, characterized only by the mutual intensity function in the aperture. The results are shown to agree with experimental results previously reported in the literature, and to be valid in and beyond the paraxial region. The results for both complete coherence and complete incoherence are recovered by using the appropriate limits for the mutual intensity function.