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Confidence sequences, anytime p-values (called p-processes in this paper),and e-processes all enable sequential inference for composite and nonparametricclasses of distributions at arbitrary stopping times. Examining the literature,one finds that at the heart of all these (quite different) approaches has beenthe identification of nonnegative (super)martingales. Thus, informally,nonnegative (super)martingales are known to be sufficient for\emph{anytime-valid} sequential inference, even in composite and nonparametricsettings. Our central contribution is to show that nonnegative martingales arealso universal -- after appropriately defining \emph{admissibility}, we showthat all admissible constructions of confidence sequences, p-processes, ore-processes must necessarily utilize nonnegative martingales. Our proofsutilize several modern mathematical tools for composite testing and estimationproblems: max-martingales, Snell envelopes, transfinite induction, and newDoob-L\'evy martingales make appearances in previously unencountered ways.Informally, if one wishes to perform anytime-valid sequential inference, thenany existing approach can be recovered or dominated using nonnegativemartingales. We provide several nontrivial examples, with special focus ontesting symmetry, where our new constructions render past methods inadmissible.We also prove the subGaussian supermartingale to be admissible.

Admissible anytime-valid sequential inference must rely on nonnegative martingales