Abstract: Evidence for the light-by-light scattering process, ????? , 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 ?b-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 |??| < 2.4 , diphoton invariant mass m?? > 5 GeV , diphoton transverse momentum pT?? < 1 GeV , and diphoton acoplanarity below 0.01. After all selection criteria are applied, 14 events are observed, compared to expectations of 9.0 ± 0.9 (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 3.7 standard deviations, and has properties consistent with those expected for the light-by-light scattering signal. The measured fiducial light-by-light scattering cross section, ?fid(?????) = 120 ± 46 (stat) ± 28 (syst) ± 12 (theo )nb , is consistent with the standard model prediction. The m?? distribution is used to set new exclusion limits on the production of pseudoscalar axion-like particles, via the ?? ? a ? ?? process, in the mass range ma = 5-90 GeV.