The main purpose of this thesis is to study quantum-electrodynamics (QED) in the presence of external background fields. We address this purpose by computing the Delbrück scattering amplitudes in the low-energy limit, the low-energy N-photon amplitudes in the presence of a constant field, the low-energy four-photon amplitudes in the presence of a constant magnetic field, the forward Compton scattering amplitudes in a constant magnetic field and the one-loop vertex correction in an arbitrary plane-wave field. In most cases, except for the vertex correction, we employ the worldline formalism to perform all calculations simultaneously for both scalar and spinor QED. We utilize the previously obtained result of the off-shell four-photon amplitude with two low- energy photons to calculate the circularly polarized amplitudes for the leading-order contributions to Delbrück scattering, assuming that the incoming and outgoing photons have low-energy. We compute the one-loop N-photon amplitudes in a constant background field considering off- shell low-energy photons in various field configurations. Assuming parallel magnetic and electric components of the background field enables us to obtain compact representations for these amplitudes involving only simple algebra and a single global proper-time integral with trigonometric integrands. Similarly, assuming a constant crossed field, we derive compact expressions for these amplitudes, represented by a single proper-time integral. The outcome of this integral, for fixed parameters, takes the form of a factorial function. The latter case is further refined by employing the spinor helicity formalism, where the helicity components are expressed solely in terms of Bernoulli numbers and spinor products. Moreover, for an arbitrary constant field, we obtain another representation of these amplitudes as series expansions in the external field. As an application, we compute the one-loop four-photon amplitudes in the presence of a pure magnetic field for off-shell low-energy photons. Using these results, we calculate the polarized amplitudes for linear and circular polarizations in two distinct scenarios: when the magnetic field is coplanar with the scattering plane and when it is orthogonal to it. We study the polarization flip of a photon scattered by an off-shell particle in the presence of a magnetic field. Specifically, we compute the Compton scattering amplitudes in a magnetic background field for off-shell massive particles and on-shell photons under the assumption that the scattering occurs in the forward direction, aligned along the same axis as the magnetic field. Additionally, we consider the polarization of the external photons to be perpendicular to each other. We apply the operator technique within the Furry picture (Volkov states) to compute the general expression of the one-loop vertex correction in an arbitrary plane-wave background field for the case of two on-shell external electrons and an off-shell external photon. We show that the ultraviolet divergence can be renormalized exactly as in vacuum while the infrared divergence is avoided by introducing a finite photon mass. This calculation completes the study of QED in a plane-wave background field at one-loop order. In most cases, except for the Delbrück scattering amplitudes, we perform non-perturbative calculations, given that the external background fields are taken into account exactly.