  General Relativity assumes that spacetime is fully described by the metric
alone. An alternative is the so called Palatini formalism where the metric and
the connections are taken as independent quantities. The metric-affine theory
of gravity has attracted considerable attention recently, since it was shown
that within this framework some cosmological models, based on some generalized
gravitational actions, can account for the current accelerated expansion of the
universe. However we think that metric-affine gravity deserves much more
attention than that related to cosmological applications and so we consider
here metric-affine gravity theories in which the gravitational action is a
general function of the scalar curvature while the matter action is allowed to
depend also on the connection which is not {\em a priori} symmetric. This
general treatment will allow us to address several open issues such as: the
relation between metric-affine $f(R)$ gravity and General Relativity (in vacuum
as well as in the presence of matter), the implications of the dependence (or
independence) of the matter action on the connections, the origin and role of
torsion and the viability of the minimal-coupling principle.
