Extended Theories of Gravity (ETGs) aim to address shortcomings of General Relativity (GR) at both infrared (IR) and ultraviolet (UV) scales. They preserve GR's successes while tackling issues like dark energy, dark matter, inflation, and quantum gravity. The review discusses principles of gravitational theories, focusing on $ f(R) $-gravity and scalar-tensor gravity in metric and Palatini approaches, as well as torsion. It explores dynamical and conformal equivalences, viability criteria, and challenges like neutrino oscillations and gravitational waves. The paper also covers the post-Newtonian and post-Minkowskian limits, the Hamiltonian formulation, and the initial value problem. Applications include spherical symmetry, post-Newtonian limits, and post-Minkowski limits. The review highlights the need for a unified framework to describe gravity, emphasizing the importance of testing ETGs against observational data. It discusses the historical development of gravity theories, from Newton to Einstein, and the challenges in unifying GR with quantum field theory. The paper addresses the quantum gravity problem, the need for a consistent theory, and the role of extended theories in explaining dark matter and dark energy. It also explores the implications of quantum gravity at high energies and the difficulties in quantizing gravity, including the non-renormalizability of gravity and the need for alternative approaches like string theory and supergravity. The review concludes that ETGs offer a promising avenue for addressing the unresolved issues in gravitational physics.Extended Theories of Gravity (ETGs) aim to address shortcomings of General Relativity (GR) at both infrared (IR) and ultraviolet (UV) scales. They preserve GR's successes while tackling issues like dark energy, dark matter, inflation, and quantum gravity. The review discusses principles of gravitational theories, focusing on $ f(R) $-gravity and scalar-tensor gravity in metric and Palatini approaches, as well as torsion. It explores dynamical and conformal equivalences, viability criteria, and challenges like neutrino oscillations and gravitational waves. The paper also covers the post-Newtonian and post-Minkowskian limits, the Hamiltonian formulation, and the initial value problem. Applications include spherical symmetry, post-Newtonian limits, and post-Minkowski limits. The review highlights the need for a unified framework to describe gravity, emphasizing the importance of testing ETGs against observational data. It discusses the historical development of gravity theories, from Newton to Einstein, and the challenges in unifying GR with quantum field theory. The paper addresses the quantum gravity problem, the need for a consistent theory, and the role of extended theories in explaining dark matter and dark energy. It also explores the implications of quantum gravity at high energies and the difficulties in quantizing gravity, including the non-renormalizability of gravity and the need for alternative approaches like string theory and supergravity. The review concludes that ETGs offer a promising avenue for addressing the unresolved issues in gravitational physics.