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I will motivate and discuss the notion of dispersive sum rules in conformal field theory. They are sum rules satisfied by the OPE data of a four-point function as a consequence of conformal dispersion relations. Physically, they are a detailed manifestation of causality. The sum rules automatically suppress double-twist operators and therefore are ideally suited for implementing analytic bootstrap with rigorously bounded errors. In theories with a large N and large gap, the sum rules provide a direct link between bulk effective field theory and its UV completion, thus constraining bulk EFT from UV completeness. In some cases, the sum rules give rise to extremal functionals, i.e. they are an analytic explanation for optimal bounds coming from the numerical bootstrap. The talk will be based on https://arxiv.org/pdf/2008.04931.pdf.