This article reviews the applications of $f(R)$ theories to various aspects of cosmology and gravity, including inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. It discusses methods to distinguish $f(R)$ theories from General Relativity observationally and experimentally, and explores extensions to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity. The authors also address models that can satisfy both cosmological and local gravity constraints, and provide a detailed discussion on the structure of relativistic stars in $f(R)$ gravity. The review is organized into several sections, covering field equations, inflationary dynamics, dark energy models, local gravity constraints, cosmological perturbations, and observational signatures of dark energy models. It concludes with a discussion on other aspects of $f(R)$ theories and modified gravity, and provides a list of references for further reading.This article reviews the applications of $f(R)$ theories to various aspects of cosmology and gravity, including inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. It discusses methods to distinguish $f(R)$ theories from General Relativity observationally and experimentally, and explores extensions to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity. The authors also address models that can satisfy both cosmological and local gravity constraints, and provide a detailed discussion on the structure of relativistic stars in $f(R)$ gravity. The review is organized into several sections, covering field equations, inflationary dynamics, dark energy models, local gravity constraints, cosmological perturbations, and observational signatures of dark energy models. It concludes with a discussion on other aspects of $f(R)$ theories and modified gravity, and provides a list of references for further reading.