  Extended Theories of Gravity can be considered a new paradigm to cure
shortcomings of General Relativity at infrared and ultraviolet scales. They are
an approach that, by preserving the undoubtedly positive results of Einstein's
Theory, is aimed to address conceptual and experimental problems recently
emerged in Astrophysics, Cosmology and High Energy Physics. In particular, the
goal is to encompass, in a self-consistent scheme, problems like Inflation,
Dark Energy, Dark Matter, Large Scale Structure and, first of all, to give at
least an effective description of Quantum Gravity. We review the basic
principles that any gravitational theory has to follow. The geometrical
interpretation is discussed in a broad perspective in order to highlight the
basic assumptions of General Relativity and its possible extensions in the
general framework of gauge theories. Principles of such modifications are
presented, focusing on specific classes of theories like f (R)-gravity and
scalar-tensor gravity in the metric and Palatini approaches. The special role
of torsion is also discussed. The conceptual features of these theories are
fully explored and attention is payed to the issues of dynamical and conformal
equivalence between them considering also the initial value problem. A number
of viability criteria are presented considering the post-Newtonian and the
post-Minkowskian limits. In particular, we discuss the problems of neutrino
oscillations and gravitational waves in Extended Gravity. Finally, future
perspectives of Extended Gravity are considered with possibility to go beyond a
trial and error approach.
