Three searches are presented for signatures of physics beyond the standard model (SM) in $\tau\tau$ final states in proton-proton collisions at the LHC, using a data sample collected with the CMS detector at $\sqrt{s} =$ 13 TeV, corresponding to an integrated luminosity of 138 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 new boson $\phi$, in addition to the H(125) boson, via gluon fusion (gg$\phi$) or in association with b quarks, ranging from ${\mathcal{O}}$(10 pb) for a mass of 60 GeV to 0.3 fb for a mass of 3.5 TeV each. The data reveal two excesses for gg$\phi$ production with local $p$-values equivalent to about three standard deviations at ${m_{\phi}} =$ 0.1 and 1.2 TeV. In a search for $t$-channel exchange of a vector leptoquark U$_{1}$, 95% CL upper limits are set on the dimensionless U$_{1}$ leptoquark coupling to quarks and $\tau$ leptons ranging from 1 for a mass of 1 TeV to 6 for a mass of 5 TeV, depending on the scenario. In the interpretations of the ${M_{\mathrm{h}}^{125}}$ and {M_{\mathrm{h},\,\text{EFT}}^{125}} minimal supersymmetric SM benchmark scenarios, additional Higgs bosons with masses below 350 GeV are excluded at 95% CL.

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