Despite the notorious achievements of General Relativity, Einstein's theory is under scrutiny due to the lack of a suitable scheme to quantize gravity as well as for the puzzling features it shows both at strong (early universe, black holes) and weak (Dark Energy problem) regime. The proposal to extend the classical theory of gravity harbours the intriguing goals to cure some of these inconsistencies. A large class of modi cations of General Relativity (GR) has been widely explored in the past; in principle, the main motivation for such early e orts was to solve the problem of non-renormalizability by providing a new framework in which (thanks to higher order corrections in the gravitational action) gravity could be quantized. The analysis of the cosmological implications of such models also showed a number of peculiar features that justi ed further developments. The ultraviolet modi cations that naturally arise at high energy in the context of quantum gravity have been taken into account for their impact on the phenomenology of the very early universe. Furthermore, it was recently argued that alternative infrared extensions of the Einstein-Hilbert (EH) action could be invoked to presumably alleviate the Dark Sector problem.
