The off-shell vector-tensor multiplet is considered in an arbitrary ba
ckground of N = 2 vector supermultiplets. We establish the existence o
f two inequivalent versions, characterized by different Chem-Simons co
uplings. In one version the vector field of the vector-tensor multiple
t is contained quadratically in the Chem-Simons term, which implies no
n-linear terms in the supersymmetry transformations and equations of m
otion. In the second version, which requires a background of at least
two abelian vector supermultiplets, the supersymmetry transformations
remain at most linear in the vector-tensor components. This version is
of the type known to arise from reduction of tensor supermultiplets i
n six dimensions. Our work applies to any number of vector-tensor mult
iplets. (C) 1997 Elsevier Science B.V.