Evidence for the light-by-light scattering process, γγ→γγ, in ultraperipheral PbPb collisions at a centre-of-mass energy per nucleon pair of 5.02TeV is reported. The analysis is conducted using a data sample corresponding to an integrated luminosity of 390μb−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 ETγ>2GeV, pseudorapidity |ηγ|<2.4, diphoton invariant mass mγγ>5GeV, diphoton transverse momentum pTγγ<1GeV, 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, σfid(γγ→γγ)=120±46(stat)±28(syst)±12(theo)nb, is consistent with the standard model prediction. The mγγ distribution is used to set new exclusion limits on the production of pseudoscalar axion-like particles, via the Image 1 process, in the mass range Image 2.