Extended Theories of Gravity (ETGs) are proposed as a new paradigm to address the shortcomings of General Relativity (GR) at both infrared and ultraviolet scales. These theories aim to incorporate recent conceptual and experimental issues in Astrophysics, Cosmology, and High Energy Physics, such as inflation, dark energy, dark matter, large-scale structure, and quantum gravity. The review covers the basic principles of gravitational theories, the geometrical interpretation of GR, and the possible extensions within the framework of gauge theories. Specific theories like f(R)-gravity and scalar-tensor gravity are discussed, along with their equivalence and the role of torsion. The Hamiltonian formulation, initial value problem, and viability criteria in post-Newtonian and post-Minkowskian limits are also explored. The article concludes with a discussion on future perspectives and the need for a more comprehensive approach to gravity.Extended Theories of Gravity (ETGs) are proposed as a new paradigm to address the shortcomings of General Relativity (GR) at both infrared and ultraviolet scales. These theories aim to incorporate recent conceptual and experimental issues in Astrophysics, Cosmology, and High Energy Physics, such as inflation, dark energy, dark matter, large-scale structure, and quantum gravity. The review covers the basic principles of gravitational theories, the geometrical interpretation of GR, and the possible extensions within the framework of gauge theories. Specific theories like f(R)-gravity and scalar-tensor gravity are discussed, along with their equivalence and the role of torsion. The Hamiltonian formulation, initial value problem, and viability criteria in post-Newtonian and post-Minkowskian limits are also explored. The article concludes with a discussion on future perspectives and the need for a more comprehensive approach to gravity.