This book is about the application of the nonperturbative methods using Schwinger-Dyson equations (SDEs) to study gauge covariance of the fermion-photon vertex in the case of quenched, massless, chirally symmetric quantum electrodynamics (QED). We consider the problem of designing a fermion-photon vertex ansatz to maintain as closely as possible gauge covariance of the SDEs, with three dimensional QED as a test case. Based on the so called transverse condition, we construct a class of vertex ansatze which depends on a single parameter. We use the photon polarization scalar as a test of the gauge covariance of the vertex ansatz. A viable vertex ansatz is obtained, giving a polarization scalar which is close to the one in Landau gauge. The transverse condition itself has been questioned by several people, who claim that the fermion-photon vertex should retain its transverse part in Landau gauge. We explore briefly of how this assertion may affect the result by obtaining the solution of quenched, massless, chirally symmetric fermion SDE in arbitrary dimensional QED, and comparing its expansion with the result in perturbation theory up to one-loop order.