In central collisions at relativistic heavy ion colliders like the Relativistic Heavy Ion Collider RHIC/Brookhaven and the Large Hadron Collider LHC (in its heavy ion mode) at CERN/Geneva, one aims at detecting a new form of hadronic matter - the Quark Gluon Plasma. It is the purpose of this review to discuss a complementary aspect of these collisions, the very peripheral ones. Due to coherence, there are strong electromagnetic fields of short duration in such collisions. They give rise to

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... n-photon and photon-nucleus collisions with high flux up to an invariant mass region hitherto unexplored experimentally. After a general survey photon-photon luminosities in relativistic heavy ion collisions are discussed. Special care is taken to include the effects of strong interactions and nuclear size. Then photon-photon physics at various gamma-gamma-invariant mass scales is discussed. The region of several GeV, relevant for RHIC is dominated by QCD phenomena (meson and vector meson pair production). Invariant masses of up to about 100 GeV can be reached at LHC, and the potential for new physics is discussed. Photonuclear reactions and other important background effects, especially diffractive processes are also discussed. A special chapter is devoted to lepton-pair production, especially electron-positron pair production; due to the strong fields new phenomena, especially multiple e+-e- pair production, will occur there.