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We review the recent literature on modified theories of gravity in thePalatini approach. After discussing the motivations that lead to consideralternatives to Einstein's theory and to treat the metric and the connection asindependent objects, we review several topics that have been recently studiedwithin this framework. In particular, we provide an in-depth analysis of thecosmic speedup problem, laboratory and solar systems tests, the structure ofstellar objects, the Cauchy problem, and bouncing cosmologies. We also discussthe importance of going beyond the f(R) models to capture otherphenomenological aspects related with dark matter/energy and quantum gravity.

Palatini Approach to Modified Gravity: f(R) Theories and Beyond