  This is a brief review of modified gravity cosmologies. Generically
extensions of gravity action involve higher derivative terms, which can result
in ghosts and instabilities. There are three ways to circumvent this:
Chern-Simons terms, first order variational principle and nonlocality. We
consider recent cosmological applications of these three classes of modified
gravity models, in particular to the dark energy problem. The viable parameter
spaces can be very efficiently constrained by taking into account cosmological
data from all epochs in addition to Solar system tests and stability
considerations. We make some new remarks concerning so called algebraic
scalar-tensor theories, biscalar reformulation of nonlocal actions involving
the inverse d'Alembertian, and a possible covariant formulation holographic
cosmology with nonperturbative gravity.
