Cosmologies with varying light speed

We analyze a generalization of general relativity that incorporates a cosmic time variation of the velocity of light in vacuum, c, and the Newtonian gravitation ``constant'' G proposed by Albrecht and Magueijo. We find exact solutions for Friedmann universes and determine the rate of variation of c required to solve the flatness and classical cosmological constant problems. Potential problems with this approach to the resolution of the flatness and classical cosmological constant problems are highlighted. Reformulations are suggested which give the theory a more desirable limit as a theory of varying G in the limit of constant c and its relationship to theories with varying electron charge and constant c are discussed.