We investigate the dynamical generation of fermion mass in quantum electrodynamics (QED). This non-perturbative study is performed using a truncated set of Schwinger-Dyson equations for the fermion and the photon propagator. First, we study dynamical fermion mass generation in quenched QED with the Curtis-Pennington vertex, which satisfies the Ward-Takahashi identity and moreover ensures the multiplicative renormalizability of the fermion propagator. We apply bifurcation analysis to determine the critical point for a general covariant gauge. In the second part of this work we investigate the dynamical generation of fermion mass in full, unquenched QED. We develop a numerical method to solve the system of three coupled non-linear equations for the dynamical fermion mass, the fermion wavefunction renormalization and the photon renormalization function. Much care is taken to ensure the high accuracy of the solutions. Moreover, we discuss in detail the proper numerical cancellation of the quadratic divergence in the vacuum polarization integral and the requirement of using smooth approximations to the solutions. To achieve this, we improve the numerical method by introducing the Chebyshev expansion method. We apply this method to the bare vertex approximation to unquenched QED to determine the critical coupling for a variety of approximations. This culminates in the detailed, highly accurate, solution of the Schwinger-Dyson equations for dynamical fermion mass generation in QED including both, the photon renormalization function and the fermion wavefunction renormalization in a consistent way, in the bare vertex approximation and, for the first time, using improved vertices. We introduce new improvements to the numerical method, to achieve the accuracy necessary to avoid unphysical quadratic divergences in the vacuum polarization with the Ball-Chiu vertex.