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

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