  The way one chooses to couple gravity to matter is an essential
characteristic of any gravitational theory. In theories where the gravitational
field is allowed to have more degrees of freedom than those of General
Relativity (e.g. scalar-tensor theory, f(R) gravity) this issue often becomes
even more important. We concentrate here on f(R) gravity treated within the
Palatini variational principle and discuss how the coupling between matter and
the extra degrees of freedom of gravity (the independent connections in our
case) affects not only the resulting phenomenology but even the geometrical
meaning of fundamental fields.
