Renormalization Group (RG) is a powerful method for investigating quantum effects of matter fields on curved background. The formalism of RG in curved space is well known since 1984, but its applications to cosmology and black hole physics require more knowledge and opens a new interesting field of study. We review recent results about the derivation of renormalization group in a mass-dependent scheme and also consider ambiguities of conformal anomaly using dimensional and covariant Pauli-Villars regularizations.
