Cosmology in Palatini theories of gravity

We discuss recent results on the cosmology of extended theories of gravity formulated in the Palatini approach, i.e., assuming that metric and connection are independent fields. In particular, we focus on the attempts to explain the cosmic speedup with f(R) theories and on models that avoid the big bang singularity. The field equations for gravity Lagrangians of the form f(g_{\mu\nu},{R^\alpha}_{\beta\mu\nu}) (including torsion) are explicitly derived and discussed.