  Anomalies of discrete R-symmetries appearing in heterotic orbifold models are
studied. We find that the mixed anomalies for different gauge groups satisfy
the universal Green-Schwarz (GS) condition, indicating that these anomalies are
canceled by the GS mechanism. An exact relation between the anomaly
coefficients of the discrete R-symmetries and one-loop beta-function
coefficients is obtained. We also find that the discrete R-symmetries have a
good chance to be unbroken down to the supersymmetry breaking scale. Even below
this scale a $Z_2$ subgroup is unbroken, which may be an origin of the R-parity
of the minimal supersymmetric standard model. Relations between the R-symmetry
anomalies and T-duality anomalies are also investigated.
