  We propose a harmonic superspace description of the non-linear vector-tensor
N=2 multiplet. We show that there exist two inequivalent version: the old one
in which one of the vectors is the field-strength of a gauge two-form, and a
new one in which this vector satisfies a non-linear constraint and cannot be
expressed in terms of a potential. In this the new version resembles the
non-linear N=2 multiplet. We perform the dualization of both non-linear
versions and discuss the resulting K\"ahler potentials. Finally, we couple the
non-linear vector-tensor multiplet to an abelian background gauge multiplet.
