This review provides a comprehensive overview of modified gravity theories in cosmology, focusing on inflation, bouncing cosmology, and late-time acceleration. The authors systematically discuss various forms of modified gravity, including $F(R)$, $F(G)$, and $F(T)$ theories, as well as alternative proposals. They emphasize the formalism of these theories and their potential to describe the early and late-time evolution of the universe. Key topics include: 1. **Inflationary Dynamics**: The review covers the inflationary era, presenting scalar field descriptions, $f(\varphi, R)$ theories, and the inflation from Gauss-Bonnet and $F(T)$ theories. It also discusses singular inflation scenarios and the graceful exit from inflation. 2. **Bouncing Cosmologies**: The authors explore bouncing cosmologies, which describe a universe that contracts and then expands again, providing a non-singular alternative to the Big Bang. They discuss the stability of these models and the observational signatures that may distinguish them from standard inflation. 3. **Dark Energy**: The review addresses the dark energy era, showing how $F(R)$ and $f(\mathcal{G})$ gravity can explain the late-time acceleration of the universe. It also discusses the unification of inflation and dark energy, the phantom cosmological evolution, and the impact of dark energy oscillations. 4. **Astrophysical Applications**: The authors present astrophysical solutions in modified gravity, such as neutron stars, quark stars, black holes, and wormholes, highlighting how modified gravity changes the conditions for these solutions. 5. **Stability and Viability**: The review concludes with a discussion on the stability of modified gravity solutions and the conditions necessary for a viable cosmological model. It emphasizes the importance of compatibility with observational data and the need for new phenomena to distinguish modified gravity theories. The aim of this review is to provide a pedagogical and comprehensive resource for understanding the latest developments in modified gravity, making it accessible to both experts and non-experts.This review provides a comprehensive overview of modified gravity theories in cosmology, focusing on inflation, bouncing cosmology, and late-time acceleration. The authors systematically discuss various forms of modified gravity, including $F(R)$, $F(G)$, and $F(T)$ theories, as well as alternative proposals. They emphasize the formalism of these theories and their potential to describe the early and late-time evolution of the universe. Key topics include: 1. **Inflationary Dynamics**: The review covers the inflationary era, presenting scalar field descriptions, $f(\varphi, R)$ theories, and the inflation from Gauss-Bonnet and $F(T)$ theories. It also discusses singular inflation scenarios and the graceful exit from inflation. 2. **Bouncing Cosmologies**: The authors explore bouncing cosmologies, which describe a universe that contracts and then expands again, providing a non-singular alternative to the Big Bang. They discuss the stability of these models and the observational signatures that may distinguish them from standard inflation. 3. **Dark Energy**: The review addresses the dark energy era, showing how $F(R)$ and $f(\mathcal{G})$ gravity can explain the late-time acceleration of the universe. It also discusses the unification of inflation and dark energy, the phantom cosmological evolution, and the impact of dark energy oscillations. 4. **Astrophysical Applications**: The authors present astrophysical solutions in modified gravity, such as neutron stars, quark stars, black holes, and wormholes, highlighting how modified gravity changes the conditions for these solutions. 5. **Stability and Viability**: The review concludes with a discussion on the stability of modified gravity solutions and the conditions necessary for a viable cosmological model. It emphasizes the importance of compatibility with observational data and the need for new phenomena to distinguish modified gravity theories. The aim of this review is to provide a pedagogical and comprehensive resource for understanding the latest developments in modified gravity, making it accessible to both experts and non-experts.