QED in strong EM fields predicts many interesting phenomena, such as the birefringence and photon splitting due to vacuum polarization and the Schwinger production of charged pairs or photons. Recently the vacuum polarization effect of magnetars has been observed. The Schwinger pair production has been known for long as a possible origin of some cosmic rays.
On the other hand, QED in curved spacetimes has both QED effect and curvature effect. We review two field theoretical approaches to explore QED vacuum polarization: perturbative approach to effective actions and nonperturbative approach to the exact one-loop effective actions. The QED action in general curved spacetimes may be perturbatively calculated by employing the string-inspired worldline formalism. Then, the vacuum polarization predicts phenomena, such as the superluminal photon propagation and the violation of Lorentz invariance and CTP, and applies to black holes and cosmology.