  Scale invariance is considered in the context of gravitational theories where
the action, in the first order formalism, is of the form $S = \int L_{1} \Phi
d^4x$ + $\int L_{2}\sqrt{-g}d^4x$ where the volume element $\Phi d^4x$ is
independent of the metric. For global scale invariance, a "dilaton" $\phi$ has
to be introduced, with non-trivial potentials $V(\phi)$ = $f_{1}e^{\alpha\phi}$
in $L_1$ and $U(\phi)$ = $f_{2}e^{2\alpha\phi}$ in $L_2$. This leads to
non-trivial mass generation and a potential for $\phi$ which is interesting for
inflation. Interpolating models for natural transition from inflation to a
slowly accelerated universe at late times appear naturally. This is also
achieved for "Quintessential models", which are scale invariant but formulated
with the use of volume element $\Phi d^4x$ alone. For closed strings and branes
(including the supersymmetric cases), the modified measure formulation is
possible and does not require the introduction of a particular scale (the
string or brane tension) from the begining but rather these appear as
integration constants.
