Tau-lepton Physics at the FCC-ee circular e$^+$e$^-$ Collider

The future FCC-ee collider is designed to deliver $\mathrm{e^+e^-}$ collisions to study with ultimate precision the Z, W, and Higgs bosons, and the top quark. In a high-statistics scan around the Z pole, $1.3\times 10^{11}$ events $\mathrm{Z}\to\tau\tau$ will be produced, the largest sample of $\tau\tau$ events foreseen at any lepton collider. With their large boost, $\tau$ leptons from Z decays are particularly well suited for precision measurements. The focus of this report is on tests of lepton universality from precision measurement of $\boldsymbol{\tau}$ properties and on tests of charged lepton flavour violation in Z decays and in $\tau$ decays. In both of these areas, FCC-ee promises sensitivities well beyond present experimental limits.