The very strong electromagnetic fields present in ultraperipheral relativistic heavy-ion collisions lead to important higher-order effects of the electromagnetic interaction. These multi-photon exchange processes are studied using perturbation theory and the sudden or Glauber approximation. In many important cases, the multi-photon amplitudes factorize into independent single-photon amplitudes. These amplitudes have a common impact parameter vector, which induces correlations between the amplitudes. Impact-parameter dependent equivalent-photon spectra for simultaneous excitation are calculated, as well as, impact-parameter dependent γγ-luminosities. Excitations, like the multi-phonon giant dipole resonances, vector meson production and multiple e+e--pair production can be treated analytically in a bosonic model, analogous to the emission of soft photons in QED.