We review the current status of nonperturbative studies of gauge field theory using the Dyson-Schwinger equation formalism and its application to hadronic physics. We begin with an introduction to the formalism and a discussion of renormalisation in this approach. We then review the current status of studies of Abelian gauge theories [e.g., strong coupling quantum electrodynamics] before turning our attention to the non-Abelian gauge theory of the strong interaction, quantum chromodynamics. We discuss confinement, dynamical chiral symmetry breaking and the application and contribution of these techniques to our understanding of the strong interactions. KEYWORDS confinement of quarks and gluons; dynamical chiral symmetry breaking; Dyson-Schwinger equations; hadrons; quantum electrodynamics; quantum chromodynamics.

We study a model Dyson-Schwinger equation for the quark propagator closed using an Ansatz for the gluon propagator of the formand two Ansätze for the quark-gluon vertex: the minimal Ball-Chiu and the modified form suggested by Curtis and Pennington. Using the quark condensate as an order parameter, we find that there is a critical value of b = b c such that the model does not support dynamical chiral symmetry breaking for b > b c . We discuss and apply a confinement test which suggests that, for all values of b, the quark propagator in the model is not confining. Together these results suggest that this Ansatz for the gluon propagator is inadequate as a model since it does not yield the expected behaviour of QCD.