A search is presented for charged Higgs bosons in the H$^{\pm}$ $\to$ $\tau^{\pm}\nu_\tau$ decay mode in the hadronic final state and in final states with an electron or muon. The search is based on proton-proton collision data recorded by the CMS experiment in 2016 at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The results agree with the background expectation from the standard model. Upper limits at 95% confidence level are set on the production cross section times branching fraction to $\tau^{\pm}\nu_\tau$ for an H$^{\pm}$ in the mass range of 80 GeV to 3 TeV, including the region near the top quark mass. The observed limit ranges from 6 pb at 80 GeV to 5 fb at 3 TeV. The limits are interpreted in the context of the minimal supersymmetric standard model $m_\mathrm{h}^\mathrm{mod-}$ scenario.