Connecting early and late universe by f(R) gravity

Inflation and dark energy are two of the most relevant aspects of modern cosmology. These different epochs provide the universe is passing through accelerated phases soon after the Big-Bang and at present stage of its evolution. In this review paper, we discuss that both eras can be, in principle, described by a geometric picture, under the standard of $f(R)$ gravity. We give the fundamental physics motivations and outline the main ingredients of $f(R)$ inflation, quintessence and cosmography. This wants to be a quick summary of $f(R)$ paradigm without claiming of completeness.