Three searches are presented for signatures of physics beyond the standard model (SM) in $\tau\tau$ final states in proton-proton collisions at the CERN LHC, using a data sample collected with the CMS detector at a centre-of-mass energy of \sqrt{s} = 13~\mathrm{TeV}, corresponding to an integrated luminosity of 138~\mathrm{fb}^{-1}. Upper limits at $95\%$ confidence level (CL) are set on the products of the branching fraction for the decay into $\tau$ leptons and the cross sections for the production of a resonance $\phi$ in addition to the observed Higgs boson via gluon fusion ($\mathrm{gg}\phi$) or in association with b quarks, ranging from \mathcal{O}(10~\mathrm{pb}) (at 60~\mathrm{GeV}) to 0.3 fb (at 3.5~\mathrm{TeV}) each. The data reveal two excesses for $\mathrm{gg}\phi$ production with local $p$-values equivalent to about three standard deviations at 0.1 and 1.2~\mathrm{TeV}. In a search for $t$-channel exchange of a vector leptoquark $\mathrm{U}_1$, $95\%$ CL upper limits are set on the $\mathrm{U}_1$ coupling to quarks and $\tau$ leptons ranging from 1 (at 1~\mathrm{TeV}) to 6 (at 5~\mathrm{TeV}), depending on the scenario. In the interpretation of minimal supersymmetric SM (MSSM) benchmark scenarios, additional Higgs bosons with masses below 350~\mathrm{GeV} are excluded at $95\%$ CL. $\textit{This document has been revised with respect to the version dated March 13th, 2022.}$

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