The aim of this paper is to give an expository account of the uniqueness theorem for cohomology theories on the category of locally compact Hausdorff spaces and proper continuous functions. The uniqueness theorem can be applied to give a proof of the known result that the Chern character induces an isomorphism K(X)\(\otimes {\mathbb{Q}}\cong \check H^{even}(X; {\mathbb{Q}})\) for every locally compact space X. The uniqueness theorem is also used to give a new proof of the duality theorem in stable homotopy theory: \(\{A;S^ n-B\}^ q\cong \{B;S^ n-A\}^ q\) where A,B are compact subsets of \(S^ n\).