  Interactions in atomic and molecular systems are dominated by electromagnetic
forces and the theoretical framework must be in the quantum regime. The
physical theory for the combination of quantum mechanics and electromagnetism,
quantum electrodynamics has been established by the mid-twentieth century,
primarily as a scattering theory. To describe atoms and molecules, it is
important to consider bound states. In the non-relativistic quantum mechanics
framework, bound states can be efficiently computed using robust and general
methodologies with systematic approximations developed for solving wave
equations. With the sight of the development of a computational quantum
electrodynamics framework for atomic and molecular matter, the field theoretic
Bethe-Salpeter wave equation expressed in space-time coordinates, its exact
equal-time variant and emergence of a relativistic wave equation is reviewed. A
computational framework, with initial applications and future challenges in
relation with precision spectroscopy, is also highlighted.
