The paper reviews the current status of nonperturbative studies of gauge field theory using the Dyson-Schwinger equation (DSE) formalism and its application to hadronic physics. It begins with an introduction to the DSE formalism and discusses renormalization in this approach. The paper then reviews the current status of studies of Abelian gauge theories, such as strong coupling quantum electrodynamics (QED), before turning to non-Abelian gauge theory, quantum chromodynamics (QCD). It discusses confinement, dynamical chiral symmetry breaking, and the application of these techniques to our understanding of the strong interactions. The DSE formalism is based on an infinite tower of coupled integral equations that relate the Green's functions of a field theory to each other. Solving these equations provides a solution of the theory, as a field theory is completely defined when all of its n-point Green's functions are known. The DSEs include the Bethe-Salpeter equation, which is needed for the description of relativistic two-body scattering and bound states. Quantitative studies of a field theory must be based on systematic approximation schemes, such as perturbation theory for weak coupling, lattice studies, large N expansions, and hbar expansions. The paper reviews the current status of DSE-based studies of QED in three and four dimensions, with particular emphasis on gauge covariance and multiplicative renormalisability. These studies are extremely useful as a guide to the more complicated case of QCD. The paper then examines the gauge boson sector of QCD, focusing on the infrared behaviour of the gluon propagator, which is thought to be crucial to confinement in QCD. It then examines the quark sector of QCD, discussing the crucial issues of dynamical chiral symmetry breaking and quark confinement. The paper reviews applications of the ideas discussed to studies of hadronic structure and concludes with a summary and discussion of possible future extensions of these studies.The paper reviews the current status of nonperturbative studies of gauge field theory using the Dyson-Schwinger equation (DSE) formalism and its application to hadronic physics. It begins with an introduction to the DSE formalism and discusses renormalization in this approach. The paper then reviews the current status of studies of Abelian gauge theories, such as strong coupling quantum electrodynamics (QED), before turning to non-Abelian gauge theory, quantum chromodynamics (QCD). It discusses confinement, dynamical chiral symmetry breaking, and the application of these techniques to our understanding of the strong interactions. The DSE formalism is based on an infinite tower of coupled integral equations that relate the Green's functions of a field theory to each other. Solving these equations provides a solution of the theory, as a field theory is completely defined when all of its n-point Green's functions are known. The DSEs include the Bethe-Salpeter equation, which is needed for the description of relativistic two-body scattering and bound states. Quantitative studies of a field theory must be based on systematic approximation schemes, such as perturbation theory for weak coupling, lattice studies, large N expansions, and hbar expansions. The paper reviews the current status of DSE-based studies of QED in three and four dimensions, with particular emphasis on gauge covariance and multiplicative renormalisability. These studies are extremely useful as a guide to the more complicated case of QCD. The paper then examines the gauge boson sector of QCD, focusing on the infrared behaviour of the gluon propagator, which is thought to be crucial to confinement in QCD. It then examines the quark sector of QCD, discussing the crucial issues of dynamical chiral symmetry breaking and quark confinement. The paper reviews applications of the ideas discussed to studies of hadronic structure and concludes with a summary and discussion of possible future extensions of these studies.