  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.
