  We discuss the role of meson exchange mechanisms in $\gamma \gamma \to \gamma
\gamma$ scattering. Several pseudoscalar ($\pi^0$, $\eta$, $\eta'(958)$,
$\eta_c(1S)$, $\eta_c(2S)$), scalar ($f_{0}(500)$, $f_0(980)$, $a_0(980)$,
$f_0(1370)$, $\chi_{c0}(1P)$) and tensor ($f_2(1270)$, $a_2(1320)$,
$f_2'(1525)$, $f_2(1565)$, $a_2(1700)$) mesons are taken into account. We
consider not only $s$-channel but also for the first time $t$- and $u$-channel
meson exchange amplitudes corrected for off-shell effects including vertex form
factors. We find that, depending on not well known vertex form factors, the
meson exchange amplitudes interfere among themselves and could interfere with
fermion-box amplitudes and modify the resulting cross sections. The meson
contributions are shown as a function of collision energy as well as angular
distributions are presented. Interesting interference effects separately for
light pseudoscalar, scalar and tensor meson groups are discussed. The meson
exchange contributions may be potentially important in the context of a
measurement performed recently in ultraperipheral collisions of heavy ions by
the ATLAS collaboration. The light-by-light interactions could be studied in
future in electron-positron collisions by the Belle II at SuperKEKB
accelerator.
