  Evidence for the light-by-light scattering process, $\gamma\gamma$ $\to$
$\gamma\gamma$, 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$_\mathrm{T}^{\gamma}$ $>$ 2 GeV, pseudorapidity
$|\eta^{\gamma}|$ $\lt$ 2.4, diphoton invariant mass $m^{\gamma\gamma}$ $\gt$ 5
GeV, diphoton transverse momentum $p_\mathrm{T}^{\gamma\gamma}$ $\lt$ 1 GeV,
and diphoton acoplanarity below 0.01. After all selection criteria are applied,
14 events are observed, compared to expectations of 9.0 $\pm$ 0.9 (theo) events
for the signal and 4.0 $\pm$ 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, $\sigma_\mathrm{fid}
(\gamma\gamma$ $\to$ $\gamma\gamma) =$ 120 $\pm$ 46 (stat) $\pm$ 28 (syst)
$\pm$ 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 $\gamma\gamma$ $\to$ a
$\to$ $\gamma\gamma$ process, in the mass range $m_{\mathrm{a}} =$ 5-90 GeV.
