We study inflation and late-time acceleration in the expansion of the universe in non-minimalelectromagnetism, in which the electromagnetic field couples to the scalar curvaturefunction. It is shown that power-law inflation can be realized due to the non-minimalgravitational coupling of the electromagnetic field, and that large-scale magnetic fields canbe generated due to the breaking of the conformal invariance of the electromagnetic fieldthrough its non-minimal gravitational coupling. Furthermore, it is demonstrated that bothinflation and the late-time acceleration of the universe can be realized in a modifiedMaxwell-F(R) gravity which is consistent with solar-system tests and cosmological bounds and free ofinstabilities. At small curvature typical for the current universe the standard Maxwelltheory is recovered. We also consider the classically equivalent form of non-minimalMaxwell-F(R) gravity, and propose the origin of the non-minimal gravitational coupling function based onrenormalization-group considerations.