  In this paper we shall show that, unless the affine geometrical structure of
the underlying spacetime manifold is specified, there is an ambiguity in the
understanding of the scale invariance -- also Weyl invariance -- of the given
theory of gravity. In this regard we cast doubts on several well-known theories
which are claimed to be scale-invariant setups. We shall show that in a
consistent scale invariant theory not only the action but also the geometrical
laws which govern the affine structure of the underlying spacetimes are
invariant under the Weyl gauge transformations.
