  The conformal anomaly for 4D gravity-matter theories, which are non-minimally
coupled with the dilaton, is systematically studied. Special care is taken for:
rescaling of fields, treatment of total derivatives, hermiticity of the system
operator and choice of measure. Scalar, spinor and vector fields are taken as
the matter quantum fields and their explicit conformal anomalies in the
gravity-dilaton background are found. The cohomology analysis is done and some
new conformal invariants and trivial terms, involving the dilaton, are
obtained. The symmetry of the constant shift of the dilaton field plays an
important role. The general structure of the conformal anomaly is examined. It
is shown that the dilaton affects the conformal anomaly characteristically for
each case: 1)[Scalar] The dilaton changes the conformal anomaly only by a new
conformal invariant, $I_4$; 2)[Spinor] The dilaton does {\it not} change the
conformal anomaly; 3)[Vector] The dilaton changes the conformal anomaly by
three new (generalized) conformal invariants, $I_4,I_2,I_{1}$. We present some
new anomaly formulae which are useful for practical calculations. Finally, the
anomaly induced action is calculated for the dilatonic Wess-Zumino model. We
point out that the coefficient of the total derivative term in the conformal
anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the
disagreement between calculations in refs.\cite{ENO,NO,SI97,KLV} and the result
of Hawking-Bousso\cite{BH}.
