PT Journal AU Olmo, GJ TI Palatini approach to modified gravity: f(R) theories and beyond SO International Journal of Modern Physics D JI Int. J. Mod. Phys. D PY 2011 BP 413 EP 462 VL 20 IS 4 DI 10.1142/S0218271811018925 LA English DE Palatini formalism; modified gravity; cosmic speed-up; dark energy; dark matter; MOND; quantum gravity phenomenology; Hamiltonian formulation; stellar structure; Cauchy problem; solar system tests AB We review the recent literature on modified theories of gravity in the Palatini approach. After discussing the motivations that lead to consider alternatives to Einstein's theory and to treat the metric and the connection as independent objects, we review several topics that have been recently studied within this framework. In particular, we provide an in-depth analysis of the cosmic speed-up problem, laboratory and solar system tests, the structure of stellar objects, the Cauchy problem, and bouncing cosmologies. We also discuss the importance of going beyond the f(R) models to capture other phenomenological aspects related with dark matter/energy and quantum gravity. ER