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We consider modified gravity which may describe the early-time inflationand/or late-time cosmic acceleration of the universe. In particular, we discussthe properties of $F(R)$, $F(G)$, string-inspired andscalar-Einstein-Gauss-Bonnet gravities, including their FRW equations and fluidor scalar-tensor description. Simplest accelerating cosmologies areinvestigated and possibility of unified description of the inflation with darkenergy is described. The cosmological reconstruction program which permits toget the requested universe evolution from modified gravity is developed. Assome extension, massive $F(R)$ bigravity which is ghost-free theory ispresented. Its scalar-tensor form turns out to be the easiest formulation. Thecosmological reconstruction method for such bigravity is presented. The unifieddescription of inflation with dark energy in $F(R)$ bigravity turns out to bepossible.

Accelerating cosmology in modified gravity: from convenient $F(R)$ or string-inspired theory to bimetric $F(R)$ gravity