In order to investigate specific aspects of bound state calculations in a non-relativistic framework, we consider the energy-levels of a massive scalar particle, which moves in an external field and interacts in addition with a massless scalar particle. The discussion includes the following topics: dimensionally regularized bound-state calculations, ultraviolet finiteness of bound-state observables and their independence of the off-mass-shell behavior of Green functions, non-renormalizable interactions, structure of the non-relativistic two-point function, power counting, and matching. 2000 Academic Press Contents I. Introduction. II. Relativistic formulation. A. The external field. B. Radiative corrections. C. Non-renormalizable interactions. III. Non-relativistic formulation: The external field. A. The scattering sector. B. Perturbation theory for bound state. C. Energy-levels. IV. Including the dynamical light field. A. Tree graphs. B. Loops. V. The two-point function. A. The two-point function at p=0. B. The two-point function at p{0. C. Comparison with the relativistic theory. VI. Power counting and matching. A. The Lagrangian, the reduction formula, and power counting. B. Matching: The vertex function at order e 2 . C. The Hamiltonian. VII. The ground state. VIII. Summary and conclusions. Appendixes. A. One-Coulomb contribution to the energy-level shift. B. Multi-Coulomb contributions to the energy-level shift. C. Use of the equation of motion. D. Connection of the resolvent with the two-point function.