Evidence for the light-by-light scattering process, γγ → γγ, in ultraperipheral PbPb collisions at a centre-of-mass energy per nucleon pair of 5.02 TeV is reported. The analysis is conducted using a data sample corresponding to an integrated luminosity of 390 μb −1 recorded by the CMS experiment at the LHC. Light-by-light scattering processes are selected in events with two photons exclusively produced, each with transverse energy ET > 2 GeV, pseudorapidity |η | < 2.4, diphoton invariant mass mγγ > 5 GeV, diphoton transverse momentum pγγ_T < 1 GeV, and diphoton acoplanarity below 0.01. After all selection criteria are applied, 14 events are observed, compared to expectations of 11.1±1.1 (theo) events for the signal and 4.0±1.2 (stat) for the background processes. The excess observed in data relative to the background-only expectation corresponds to a significance of 4.1 standard deviations, and has properties consistent with those expected for the light-by-light scattering signal. The m γγ distribution is used to set new exclusion limits on the production of pseudoscalar axion-like particles, via the γγ → a → γγ process, in the mass range m a = 5–90 GeV.