  We shall show equivalence between Palatini-$f(\calR)$ theories and
Brans-Dicke (BD) theories at the level of action principles in generic
dimension with generic matter coupling. We do that by introducing the Helmholtz
Lagrangian associated to Palatini-$f(\calR)$ theory and then performing frame
transformations in order to recover Einstein frame and Brans-Dicke frame. This
clarifies the relation among different formulations and the transformations
among different frames. Additionally, it defines a formulation {\it a l\'a
Palatini} for the Brans-Dicke theory which is dynamically equivalent to metric
BD (unlike the standard Palatini-formulation of metric BD theory which are {\it
not} dynamically equivalent). In conclusion we discuss interpretation of
extended theories of gravitation and perspectives.
