An almost contact manifold is a \(C^{\infty}\) manifold \(M^{2n+1}\) whose structural group can be reduced to U(n)\(\times 1\); equivalently M admits a tensor field \(\phi\) of type (1,1), a vector field \(\xi\) and a 1- form \(\eta\) such that \(\phi^ 2=-I+\eta \otimes \sigma\) and \(\eta (\xi)=1\). Such a manifold admits a Riemannian metric g satisfying \(g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y)\) and we refer to (\(\phi\),\(\xi\),\(\eta\),g) as an almost contact metric structure. The fundamental 2-form \(\Phi\) of the structure is defined by \(\Phi (X,Y)=g(X,\phi Y).\) Many particular types of almost contact structures have been defined and studied in the literature, e.g. cosymplectic, Sasakian, almost cosymplectic, \(\alpha\)-Kenmotsu, \(\alpha\)-Sasakian, etc. The purpose of this paper is to fit all these classes into a general system which is, in a reasonable sense, complete. The approach is to consider a vector space \(V^{2n+1}\) with an almost contact metric structure and study the representation of the group U(n)\(\times 1\) on the space \({\mathcal C}(V)\) of tensors of type (0,3) which satisfy the symmetries of the covariant derivative of the fundamental 2-form. This gives a decomposition of \({\mathcal C}(V)\) into twelve irreducible invariant components \({\mathcal C}_ i\), \(i=1,...,12\). It is possible to form \(2^{12}\) invariant subspaces and a class of almost contact metric manifolds corresponding to each. For example, \(\{\) \(0\}\) corresponds to cosymplectic manifolds, \({\mathcal C}_ 6\) to \(\alpha\)-Sasakian, \({\mathcal C}_ 2\oplus {\mathcal C}_ 9\) to almost cosymplectic, \({\mathcal C}_ 3\oplus {\mathcal C}_ 4\oplus {\mathcal C}_ 5\oplus {\mathcal C}_ 6\oplus {\mathcal C}_ 7\oplus {\mathcal C}_ 8\) to normal almost contact metric manifolds, etc. Defining relations of the classes \({\mathcal C}_ i\) in terms of the covariant derivatives of \(\Phi\) and \(\eta\) are given and many examples of various structures are given. A note added at the end of the paper points out that after their work had been completed the authors learned that this decomposition had been obtained in the unpublished doctoral dissertation of \textit{F. Bouten}, a brief announcement of which appears in the abstracts of the IX. Österr. Math. Kongr. Salzburg 1977).