  I discuss and extend several results concerning the cancellation of discrete
gauge anomalies. I show how heavy fermions do not decouple in the presence of
discrete gauge anomalies. As a consequence, in general, cancellation of
discrete gauge anomalies cannot be described merely in terms of low energy
operators involving only the light fermions. I also discuss cancellation of
discrete gauge anomalies through a discrete version of the Green-Schwarz (GS)
mechanism as well as the possibility of discrete gauge R-symmetries and their
anomalies.
  Finally, some phenomenological applications are discussed. This includes
symmetries guaranteeing absence of FCNC in two-Higgs models and generalized
matter parities stabilizing the proton in the supersymmetric standard model. In
the presence of a discrete GS mechanism or/and gauge R-symmetries, new
possibilities for anomaly free such symmetries are found.
