  To fully clarify the invariance of the classical Liouville field theory under
the Virasoro algebra, we first elucidate in detail the concept of classical
anomaly, discuss the occurrence of two symmetry algebras associated to this
problem, and provide some new formulae to compute the classical center in a
general fashion. We apply this to the study of the symmetries of the free boson
in two dimensions. Moving to Liouville, we see how this gives rise to an
energy-momentum tensor with non-tensorial conformal transformations, in flat
space, and a non-vanishing trace, in curved space. We provide a variety of
improvements of the (local) theory, that restore Weyl invariance. With explicit
computations, we show that the covariant conservation of the
Weyl-invariance-improved energy-momentum tensor is lost, in general, and relate
the chosen improvement with the corresponding subset of preserved
diffeomorphisms. The non-tensorial transformation rule of the
Weyl-invariance-improved energy-momentum tensor in curved space is explicitly
back-traced to the Virasoro center.
