Evidence for the light-by-light scattering process, gamma gamma -> gamma gamma, 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 mu 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 E-T(gamma) > 2 GeV, pseudorapidity vertical bar eta(gamma)vertical bar < 2.4, diphoton invariant mass m(gamma gamma) > 5 GeV, diphoton transverse momentum p(T)(gamma gamma) < 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, sigma(fid)(gamma gamma -> gamma gamma) = 120 +/- 46(stat) +/- 28(syst) +/- 12(theo) nb, is consistent with the standard model prediction. The m(gamma gamma) distribution is used to set new exclusion limits on the production of pseudoscalar axion-like particles, via the gamma gamma -> a -> gamma gamma process, in the mass range m(a) = 5-90 GeV. (C) 2019 The Author(s). Published by Elsevier B.V.