  The gauging of axionic, Stueckelberg-type, shift symmetries with generalized
Chern-Simons terms and the Green-Schwarz mechanism of anomaly cancellation has
recently been studied in the context of certain string compactifications and
with regard to the phenomenology of unusual variants of Z'-bosons. In this
talk, following recent work with De Rydt, Rosseel, Schmidt and Van Proeyen, I
summarize how such theories can be reconciled with global and local N=1
supersymmetry. Two simple consistency conditions are shown to encode strong
constraints on the allowed anomalies for different types of gauge groups.
