Conformal Anomaly in 4D Gravity-Matter Theories Non-minimally Coupled with Dilaton

4 dim gravity-matter theories non-minimally coupled with the dilaton are systematically studied from the viewpoint of the conformal anomaly. Special care is taken for the following points: rescaling of fields; treatment of total derivatives; hermiticity of the system operator; 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. 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 not change the conformal anomaly; 3)[Vector] The dilaton changes the conformal anomaly by four new (generalized) conformal invariants, $I_4,I_2,I_{1a},I_{1b}$. These results show the dilaton has the legitimate status as the background field in the same way as the gravity. We present some new anomaly formulae which are useful for the practical calculation. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We also point out that the coefficient of a total derivative term in the conformal anomaly for 2D dilaton coupled scalar is ambiguous, which resolves the disagreement between calculations in (hep-th/9308020,hep-th/9706009,hep-th/9707025,hep-th/9707041) and result by Hawking-Bousso (hep-th/9705236).