  Current generalizations of the classical Einstein-Hilbert Lagrangian
formulation of General Relativity are reviewed. Some alternative variational
principles are known to reproduce Einstein's gravitational equations, and
should therefore be regarded as equivalent descriptions of the same physical
model, while other variational principles ("Scalar-tensor theories" and
"Higher-derivative theories") are commonly presented as truly alternative
physical theories. Such theories, however, are also known to admit a
reformulation which is formally identical to General Relativity (with auxiliary
fields). The physical significance of this change of variables has been
questioned by several authors in recent years. Here, we investigate to which
extent purely affine, metric-affine, scalar-tensor and purely metric theories
can be regarded as physically equivalent to GR; we show that in general this
depends on which metric tensorfield is assumed to represent the true physical
space-time geometry. For purely metric theories where the Lagrangian is a
nonlinear function f(R) of the curvature scalar, we present an argument based
on the definition of the physical energy, which leads one to regard the
rescaled metric (Einstein frame) as the true physical one. As a direct
consequence, the physical content of such "alternative" models is reset to
coincide with General Relativity, and the "Nonlinear Gravity Theories" become
nothing but exotic reformulations of General Relativity in terms of unphysical
variables.
