Unified cosmic history in modified gravity: from F(R) theory to Lorentz
  non-invariant models
  Classical generalization of general relativity is considered as gravitational
alternative for unified description of the early-time inflation with late-time
cosmic acceleration. The structure and cosmological properties of number of
modified theories, including traditional $F(R)$ and Ho\v{r}ava-Lifshitz $F(R)$
gravity, scalar-tensor theory, string-inspired and Gauss-Bonnet theory,
non-local gravity, non-minimally coupled models, and power-counting
renormalizable covariant gravity are discussed. Different representations and
relations between such theories are investigated. It is shown that some
versions of above theories may be consistent with local tests and may provide
qualitatively reasonable unified description of inflation with dark energy
epoch. The cosmological reconstruction of different modified gravities is made
in great detail. It is demonstrated that eventually any given universe
evolution may be reconstructed for the theories under consideration: the
explicit reconstruction is applied to accelerating spatially-flat FRW universe.
Special attention is paid to Lagrange multiplier constrained and conventional
$F(R)$ gravities, for last theory the effective $\Lambda$CDM era and
phantom-divide crossing acceleration are obtained. The occurrence of Big Rip
and other finite-time future singularities in modified gravity is reviewed as
well as its curing via the addition of higher-derivative gravitational
invariants.
