We review the covariant formulation of quantum field theory, and in particular DeWitt's background field method, Schwinger's proper-time representations and Pauli-Villars invariant regularisation. We explain the difficulties encountered in the application of these techniques to the definition of the effective action for chiral gauge theories, and in particular the quantum mechanical breakdown of chiral gauge invariance leading to chiral anomalies, the structure of which is reviewed in detail. An exact non-perturbative proper-time representation of the full effective action for a chiral fermion is then derived, and its relationship to the Wess-Zumino term and Goldstone-Wilczek current discussed. Finally we derive local derivative expansions of the effective action in the low energy limit, and consider the relevance of decoupling to the quantisation of anomalous chiral gauge theories.
