Cosmologies with variable gravitational constant

In 1937 Dirac presented an argument, based on the socalled large dimensionless numbers, which led him to the conclusion that the Newtonian gravitational constantG changes with epoch. Towards the end of the last century Ernst Mach had given plausible arguments to link the property of inertia of matter to the large scale structure of the universe. Mach's principle also leads to cosmological models with a variable gravitational constant. Three cosmologies which predict a variableG are discussed in this paper both from theoretical and observational points of view.