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There exist several proposals to use Weizsäcker-Williams photons generated by ultrarelativistic heavy ions to produce exotic particles in γγ 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 γγ fusion they include also electromagnetic γA interactions and strong nucleon-nucleon interactions in grazing AA collisions. All these processes are characterised by low multiplicities of produced particles. γA 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 γγ events. The possibility of selecting mesons produced in γγ fusion events via different p t cut procedures is demonstrated. *

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