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