The production of a pair of τ leptons via photon-photon fusion, γ γ → τ τ , is observed for the first time in proton-proton collisions, with a significance of 5.3 standard deviations. This observation is based on a data set recorded with the CMS detector at the LHC at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 138 fb−1. Events with a pair of τ leptons produced via photon-photon fusion are selected by requiring them to be back-to-back in the azimuthal direction and to have a minimum number of charged hadrons associated with their production vertex. The τ leptons are reconstructed in their leptonic and hadronic decay modes. The measured fiducial cross section of γ γ → τ τ is σ obs fid = 12.4 − 3.1 + 3.8 fb . Constraints are set on the contributions to the anomalous magnetic moment ( a τ ) and electric dipole moments ( d τ ) of the τ lepton originating from potential effects of new physics on the γ τ τ vertex: a τ = 0.0009 − 0.0031 + 0.0032 and | d τ | < 2.9 × 10 − 17 e cm (95% confidence level), consistent with the standard model.