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In many search channels for physics beyond the Standard Model the sensitivity is limited by the knowledge of hadronic matrix elements, or the interpretation of limits depends thereon. One way to constrain such matrix elements proceeds via dispersion relations, which allow one to relate them to their singularities via the machinery of complex analysis. In the talk, I will introduce the basic strategy and then discuss recent applications, including hadronic contributions to the anomalous magnetic moment of the muon.