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These notes are a summary of lectures given at the Instituto de Astronomia ofthe Universidad Nacional Autonoma de Mexico (UNAM), the Dipartimento di Fisicaof the Universita degli studi di Salerno (Italy), and the Instituto de FisicaCorpuscular of the Universitat de Valencia (Spain) during the year 2012. Basicmathematical aspects of Palatini theories of gravity, which are constructedassuming that metric and connection are independent geometrical entities, arebriefly introduced and discussed, and followed by a detailed derivation of thefield equations of some rather general Lagrangian theories. Applications to theearly universe are considered paying special attention to the avoidance of thebig bang singularity. The analysis of non-singular cosmologies carried out inarXiv:1005.4136 [gr-qc] is extended to include several perfect fluids.

Introduction to Palatini theories of gravity and nonsingular cosmologies