  We consider matter fields conformally coupled to a background metric and
dilaton and describe in detail a quantization procedure and related
renormalization group flow that preserve Weyl invariance. Even though the
resulting effective action is Weyl-invariant, the trace anomaly is still
present, with all its physical consequences. We discuss first the case of free
matter and then extend the result to interacting matter. We also consider the
case when the metric and dilaton are dynamical and gravitons enter in the
loops.
