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Classical generalization of general relativity is considered as gravitationalalternative for unified description of the early-time inflation with late-timecosmic acceleration. The structure and cosmological properties of number ofmodified 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-countingrenormalizable covariant gravity are discussed. Different representations andrelations between such theories are investigated. It is shown that someversions of above theories may be consistent with local tests and may providequalitatively reasonable unified description of inflation with dark energyepoch. The cosmological reconstruction of different modified gravities is madein great detail. It is demonstrated that eventually any given universeevolution may be reconstructed for the theories under consideration: theexplicit 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 andphantom-divide crossing acceleration are obtained. The occurrence of Big Ripand other finite-time future singularities in modified gravity is reviewed aswell as its curing via the addition of higher-derivative gravitationalinvariants.

Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models