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We systematically review some standard issues and also the latestdevelopments of modified gravity in cosmology, emphasizing on inflation,bouncing cosmology and late-time acceleration era. Particularly, we present theformalism of standard modified gravity theory representatives, like $F(R)$,$F(\mathcal{G})$ and $F(T)$ gravity theories, but also several alternativetheoretical proposals which appeared in the literature during the last decade.We emphasize on the formalism developed for these theories and we explain howthese theories can be considered as viable descriptions for our Universe. Usingthese theories, we present how a viable inflationary era can be produced in thecontext of these theories, with the viability being justified if compatibilitywith the latest observational data is achieved. Also we demonstrate howbouncing cosmologies can actually be described by these theories. Moreover, wesystematically discuss several qualitative features of the dark energy era byusing the modified gravity formalism, and also we critically discuss how aunified description of inflation with dark energy era can be described bysolely using the modified gravity framework. Finally, we also discuss someastrophysical solutions in the context of modified gravity, and severalqualitative features of these solutions. The aim of this review is to gatherthe different modified gravity techniques and form a virtual modified gravity"toolbox", which will contain all the necessary information on inflation, darkenergy and bouncing cosmologies in the context of the various forms of modifiedgravity.

Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution