This paper studies the \(L_{\infty\omega}\)-theory of sequences of dual groups. Here, a dual group is an Abelian group of the form \(B^*=\Hom(B, \mathbb{Z})\), where \(\mathbb{Z}\) denotes the group of integers. The short sequence associated to a dual group \(A_ 0\) is defined to be the 4-sorted structure \((A_ 0, A_ 1, A_ 2, \mathbb{Z}; \sigma_ 0; \langle_ -,_ -\rangle)\) where \(\mathbb{Z}\) is the group of integers, \(A_{n+ 1}= (A_ n)^*\), \(\sigma_ 0\) is the canonical embedding of \(A_ 0\) into \(A_ 2\) and \(\langle _ -, _ -\rangle\) is the bilinear evaluation map which takes \((a, f)\) to \(f(a)\) for \(a\in A_ n\), \(f\in A_{n+ 1}\); the group operations are also part of the structure. The long sequence associated to \(A_ 0\) is defined to be the \(\omega+ 1\)-sorted structure \((A_ n (n\in \omega), \mathbb{Z}; \sigma_ n; \langle_ -,_ -\rangle)\). These structures are a natural setting in which to speak of the properties of dual groups; for example, in the \(L_{\infty\omega}\)- language of Abelian groups one cannot say that a dual group \(A_ 0\) is reflexive (that is, \(\sigma_ 0\) is an isomorphism) but one can do so in the \(L_{\infty\omega}\)-language of these structures. The main theorems in this paper characterize short and long sequences of dual groups up to \(L_{\infty\omega}\)-equivalence by simple numerical invariants. An axiomatization of the \(L_{\infty\omega}\)-theory of short sequences of dual groups is given. It is also proved that the first order theory of any nontrival (long or short) sequence of dual groups is undecidable.