For variable gravity models the strength of gravity, as measured by Newton’s “constant” or the Planck mass, depends on the value of a scalar field, the cosmon. We discuss two simple four-parameter models with a quadratic or constant cosmon potential. They are compatible with all presently available cosmological observations, including inflation. The inflaton and the scalar field of quintessence are the same cosmon field. Dark energy constitutes a small, almost constant fraction of the energy density during the radiation- and matter-dominated epochs (early dark energy). In the present epoch we witness a transition to a new dark energy-dominated epoch. Our models are free of a big bang singularity. The stability of solutions generates an arrow of time. Our picture of the Universe is unusual, with a shrinking or static scale factor, while the masses of particles increase and the size of atoms shrinks. The evolution of the Universe can be very slow for all cosmological epochs including inflation, with typical time scale 1010  yr, and in sharp contrast to the usual big bang picture. The map to the equivalent Einstein frame with constant particle masses and expanding scale factor can be singular at the big bang.