  There exist several proposals to use Weizs\"{a}cker-Williams photons
generated by ultrarelativistic heavy ions to produce exotic particles in
$\gamma\gamma$ fusion reactions. To estimate the background conditions for such
reactions we analyze various mechanisms of meson production in very peripheral
collisions of ultrarelativistic heavy ions at RHIC and LHC energies. Besides
$\gamma\gamma$ fusion they include also electromagnetic $\gamma A$ interactions
and strong nucleon-nucleon interactions in grazing $AA$ collisions. All these
processes are characterised by low multiplicities of produced particles.
$\gamma A$ and $AA$ events are simulated by corresponding Monte Carlo codes,
RELDIS and FRITIOF. In each of these processes a certain fraction of pions is
produced close to the mid-rapidity region that gives a background for the
$\gamma\gamma$ events. The possibility of selecting mesons produced in
$\gamma\gamma$ fusion events via different $p_t$ cut procedures is
demonstrated.
