The article provides a comprehensive review of $f(R)$ theories of gravity, which have gained increasing attention due to their potential to address challenges in high-energy physics, cosmology, and astrophysics. These theories, which include higher-order curvature invariants, have a long history but have seen renewed interest in recent years. The review covers various aspects of $f(R)$ theories, including their motivations, actions, field equations, theoretical aspects, equivalence with other theories, cosmological aspects and constraints, viability criteria, and astrophysical applications. The authors discuss three formalisms: metric, Palatini, and metric-affine. They present the actions and field equations for each formalism and highlight their differences. The metric formalism is the most straightforward, while the Palatini formalism treats the metric and connection as independent variables, leading to Einstein's equations. The metric-affine formalism allows for a more general connection, potentially including torsion, but requires constraints to avoid projective invariance issues. The review also explores the cosmological evolution and constraints, viability criteria, and the confrontation with particle physics and astrophysics. It discusses the implications of $f(R)$ theories for dark matter and dark energy, and provides an overview of exact solutions and relevant constraints. The article concludes with a summary and conclusions, emphasizing the importance of $f(R)$ theories as a tool for understanding the principles and limitations of modified gravity.The article provides a comprehensive review of $f(R)$ theories of gravity, which have gained increasing attention due to their potential to address challenges in high-energy physics, cosmology, and astrophysics. These theories, which include higher-order curvature invariants, have a long history but have seen renewed interest in recent years. The review covers various aspects of $f(R)$ theories, including their motivations, actions, field equations, theoretical aspects, equivalence with other theories, cosmological aspects and constraints, viability criteria, and astrophysical applications. The authors discuss three formalisms: metric, Palatini, and metric-affine. They present the actions and field equations for each formalism and highlight their differences. The metric formalism is the most straightforward, while the Palatini formalism treats the metric and connection as independent variables, leading to Einstein's equations. The metric-affine formalism allows for a more general connection, potentially including torsion, but requires constraints to avoid projective invariance issues. The review also explores the cosmological evolution and constraints, viability criteria, and the confrontation with particle physics and astrophysics. It discusses the implications of $f(R)$ theories for dark matter and dark energy, and provides an overview of exact solutions and relevant constraints. The article concludes with a summary and conclusions, emphasizing the importance of $f(R)$ theories as a tool for understanding the principles and limitations of modified gravity.